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\begin{enumerate}gin{document}
\title{Analytic classification of a class of cuspidal foliations in
$({\mathbb C}^3,0)$}
\author{Percy Fern\'{a}ndez-S\'{a}nchez}
\address[Percy Fern\'{a}ndez]{Dpto. Ciencias - Secci\'{o}n Matem\'{a}ticas, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Av. Universitaria 1801,
San Miguel, Lima 32, Peru}
\epsilonmail{[email protected]}
\author{Jorge Mozo-Fern\'{a}ndez}
\address[Jorge Mozo Fern\'{a}ndez]{Dpto. \'{A}lgebra, An\'{a}lisis Matem\'{a}tico, Geometr\'{\i}a y Topolog\'{\i}a \\
Facultad de Ciencias, Universidad de Valladolid \\
Campus Miguel Delibes\\
Paseo de Bel\'{e}n, 7\\
47011 Valladolid - Spain}
\epsilonmail{[email protected]}
\author{Hern\'{a}n Neciosup}
\address[Hern\'{a}n Neciosup]{Dpto. Ciencias - Secci\'{o}n Matem\'{a}ticas, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Av. Universitaria 1801,
San Miguel, Lima 32, Peru}
\epsilonmail{[email protected]}
\thanks{First and third authors partially supported by the Pontificia Universidad Cat\'{o}lica del Per\'{u} project VRI-DGI 2014-0025. \\ Second author partially supported by the Ministerio de Econom\'{\i}a y Competitividad from Spain, under the Project ``\'{A}lgebra y Geometr\'{\i}a en Din\'{a}mica Real y Compleja III" (Ref.: MTM2013-46337-C2-1-P)}
\date{\today}
\begin{enumerate}gin{abstract}
In this article we study the analytic classification of certain types of quasi-homogeneous cuspidal holomorphic foliations in $({\mathbb C}^3,{\bf 0})$ via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation.
\epsilonnd{abstract}
\maketitle
\section{Introduction} \label{introduccion}
The objective of this paper is to give a contribution to the analytical classification of germs of codimension one holomorphic foliations defined on an ambient space of dimension three. More generally, assume that the dimension of the ambient space is arbitrary, call it $n$.
Let ${\mathcal F}_1, {\mathcal F}_2$ be two such germs, generated by integrable 1-forms $\omegaega_1, \omegaega_2$, respectively, holomorphic in a neighbourhood of ${\mathbf 0}\in {\mathbb C}^n$. We will say that ${\mathcal F}_1, {\mathcal F}_2$ are analytically equivalent if there exists a germ of diffeomorphism $\phi:({\mathbb C}^n, {\bf 0})\to ({\mathbb C}^n, {\bf0})$ such that $\phi^*\omegaega_1\wedge\omegaega_2=0$.
Let us mention here some of the previous results concerning this subject. When $n=2$, J. Martinet and J.-P. Ramis \cite{MR1,MR2} consider simple (reduced) singularities, both in the saddle-node case and in the resonant case. Concerning non-simple singularities, consider the nilpotent case (i.e. foliations defined by a 1-form $\omegaega$ such that its dual vector field has nilpotent, non-zero, linear part), in which these foliations admit a formal normal form $d(y^2+x^n)+x^p.u(x)dy$, where $n\geq3, p\geq2$ \cite{Takens}, and $u(0)\neq0$, and in fact an analytic pre-normal form $\omegaega=d(y^2+x^n)+A(x,y)(nydx-2xdy)$ (see \cite{Cerveau-Moussu}). These foliations have been the object of study of R. Moussu \cite{R.Moussu2}, D. Cerveau \cite{Cerveau-Moussu}, R. Meziani \cite{Meziani}, M. Berthier, P. Sad \cite{BMS} and E. Str\'{o}\.{z}yna \cite{Strozyna}. It is worth to mention here other previous works that contribute to the analytic classification of holomorphic foliations, as the ones by J.-F. Mattei \cite{Mattei-modules,Mattei} and Y. Genzmer \cite{Genzmer}. Other authors focus in the topological classification, that we will not treat in this paper. Among them, let us mention, without trying to be exhaustive, works of D. Mar\'{\i}n and J.-F. Mattei \cite{MarinMattei}, and of R. Rosas \cite{Rosas}.
In dimension $n=3$, simple singularities have been studied by D. Cerveau and J. Mozo in \cite{Cerveau-Mozo}. Concerning non-simple singularities, P. Fern\'{a}ndez and J. Mozo studied in \cite{FM} the case of quasi-ordinary, cuspidal singularities. In these works, the main tool used is, either the holonomy of one of the separatrices of the foliation or, more frequently, the projective holonomy of a certain component of the exceptional divisor that appears after the reduction of the singularities. In order to apply this technique, specially in dimension higher than two, it is of great importance to study the topology of the divisor, as the holonomy group of each component is a representation of its fundamental group.
In this paper, we will apply this technique to a kind of cuspidal foliations more general than the quasi-ordinary ones studied in \cite{FM}, more precisely, admissible quasi-homogeneous foliation (see Definition \ref{def_quasi_homogeneous} in Section \ref{general}) of generalized surface type. Recall, from \cite{FM2} and \cite{FMNeciosup1}, that a germ ${\mathcal F}$ of non dicritical foliation in $({\mathbb C}^3,{\bf 0})$, generated by an integrable 1-form $\omegaega$, is a generalized surface if every generically transverse plane section $\varphi:({\mathbb C}^2,{\bf 0})\to ({\mathbb C}^3,{\bf 0})$ (i.e., $\varphi^{*}\omegaega\not\epsilonquiv0 $) defines a generalized curve in the sense of \cite{CLS}. In particular, for generalized surfaces, once its set of separatrices has been reduced, the foliation has only simple singularities, as shown in \cite{FM2} and generalized in \cite{FMNeciosup1} for higher dimensions.
In Section \ref{general}, a cuspidal foliation in $({\mathbb C}^3,{\bf 0})$ will be defined as a generalized surface ${\mathcal F}$ having a separatrix written as $z^2+\varphi(x,y)=0$, in appropriate coordinates. We will consider in this paper the situation in which the separatrix is quasi-homogeneous. The first point would be to find an analytic normal form for these type of foliations, in the style of the ones considered by D. Cerveau and R. Moussu in dimension two. This is the main problem studied in \cite{FMNeciosup2}. In that paper, it is shown that a cuspidal, quasi-homogeneous foliation ${\mathcal F}$ with a separatrix $z^2+\varphi(x,y)=0$, can be generated by an integrable 1-form
$$\omegaega=d(z^2+\varphi)+G(\Psi,z)\cdot z\cdot\Psi\cdot\left(2{dz\over z}-{d\varphi\over\varphi}\right),$$
where $\Psi^r=\varphi, \Psi$ is not a power, and $G$ is a germ of holomorphic function in two variables. This is the starting point of the present paper.
Also in Section \ref{general}, the main definitions of the paper are presented. We will work with a concrete reduction of singularities of cuspidal quasi-homogeneous foliations, obtained via the Weierstrass-Jung method. For the sake of completeness, this reduction will be sketched in Section \ref{section3}. In Section \ref{section_toplogy_of_the_divisor}, the topology of the components of the exceptional divisor is studied. As we said before, this is important in order to identify the component of the exceptional divisor where the holonomy is computed. In particular there is one component such that, once removed the singular locus, it is homeomorphic to $({\mathbb C}^*\times{\mathbb C} )\smallsetminus\mathcal{C}$, where $\mathcal{\mathcal{C}}$ is a certain algebraic plane curve, whose topology will be interesting for us. The fundamental group will be computed in this section.
In order to lift the holonomy of a component of the divisor, a fibration transverse to the foliation outside the separatrices, is constructed in Section \ref{section_holonomy_of_the_essential_component}. Finally, Section \ref{section_cladification}, is devoted to study the main problem of the paper: to classify analytically quasi-homogeneous generalized surfaces following this method. Unfortunately, the answer is not complete, and we must impose some conditions (properties $\wp_1, \wp_2$ of Definition \ref{defini_propiedades_p_1andp_2}), that guarantee the linearization of the holonomy over some ``end component'' of the divisor, that we will call \textit{special components}. Under these conditions, Theorem \ref{thm_clasificacion_analitica} of this section is the main result of the paper, and gives the analytic classification.
As a future work, we would like to study a wider class of foliations. The existing works of Mattei and Genzmer \cite{Mattei-modules, Mattei,Genzmer} in dimension two lead us to think that different techniques should be developed, as for instance, techniques concerning deformations of foliations. We will not follow this method here.
\section{Generalities} \label{general}
In dimension $n\geq2$, a germ of codimension one holomorphic foliation ${\mathcal F}$ is called \textit{cuspidal} if, besides being generalized hypersurface (see Section \ref{introduccion} and \cite{FM2}), its separatrix is defined in appropriate coordinates $(\mathbf{x}, z)$, where $\textbf{x}:=(x_1,x_2,\cdots,x_{n-1})$, by a reduced equation $z^2+\varphi(\textbf{x})=0$, with $\nu_0 (\varphi (\textbf{x}))\geq 2$. In \cite[Thm. 1.4]{FMNeciosup2} we constructed a pre-normal form for germs of holomorphic foliation of cuspidal type. In particular, if $z^2+\varphi(\textbf{x})=0$ is a quasi-homogeneous hypersurface, we know that there exist analytic coordinates such that a generator of ${\mathcal F}$ is the 1-form
$$
\omega = d(z^2+\varphi) + G(\Psi, z) (z\cdot \Psi)\cdot \left(
2\frac{dz}{z}-\frac{d\varphi}{\varphi}\right) ,
$$
where $\varphi= \Psi^r$ for some $r\in{\mathbb N},\;\Psi$ is not a power, and $G$ is a germ of holomorphic function in two variables (\cite[Cor. 4.2]{FMNeciosup2}). \\
In dimension three, there exist analytic coordinates $(x,y,z)$ such that
$$\Psi=x^{n_1}y^{n_2}\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i},$$
where $n_i,p,q\in\mathbb{N}$, and $a_i\in {\mathbb C}^*$ are pairwise distinct. After \cite[Thm. 2.4]{FMNeciosup2} and \cite[Thm. 11]{FM2}, the reduction of the singularities of this type of foliations follows the same scheme that the desingularization of their separatrix $S$,
$$S:=z^2+x^{n_1}y^{n_2}\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0.$$
A key to prove the main result of this paper (Theorem \ref{thm_clasificacion_analitica}) is the existence of the first integral in the intersections of the components of the exceptional divisor.
Let us mention in dimension two the work \cite{D.Marin-Thesis}, where it is said that the classification technique from R. Moussu \cite{R.Moussu2} and F. Loray \cite{Loray-Thesis} can be used in the quasi-homogeneous framework, \textit{provided that the coordinate axis are not part of this set of separatrices}. i.e., after reduction of singularities, all the separatrices lie on the same component of the exceptional divisor. Because of this, we will assume an analogous hypothesis in dimension three, i.e., $n_1=n_2=0$,
and Theorem \ref{thm_clasificacion_analitica} will be proved based on this property. This means that coordinate hyperplanes are not part of its set of separatrices. Let us mention that on the other hand, in \cite{Genzmer} this hypothesis is not considered, but other techniques of a rather different nature have been used, which will not be addressed here. According to this, let us state the following definition:
\begin{enumerate}gin{defin}\label{def_quasi_homogeneous}
A cuspidal holomorphic foliation, ${\mathcal F}$, in $({\mathbb C}^3,{\bf 0})$ is called {\bf quasi-homogeneous of admissible type}, if its separatrix, in appropriate coordinates, is defined by the equation
$$S=z^2+\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0,$$
where $p,q\geq2,\;a_i\in{\mathbb C}^*$, and $a_i\neq a_j,$ if $i\neq j$.
\epsilonnd{defin}
So, a quasi-homogeneous foliations of admissible type is generated by a 1-form
\begin{enumerate}gin{equation}\label{forma_quasi_homogeneous}
\Omegaega=d(z^2+\varphi)+G(\Psi,z)\cdot z\cdot\Psi\left({d\varphi\over\varphi}-2{dz\over z}\right),
\epsilonnd{equation}
where $\varphi=\Psi^r=\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i},\;p,q\geq2,\; r=\gcd (d_1,\cdots,d_l)$.
In general, let us denote by ${\mathcal F}_{\omegaega}$ a holomorphic foliation of codimension one generated by an integrable 1-form $\omegaega\in\Omegaega^1({\mathbb C}^n,{\bf 0})$.
Not every such 1-form defines a quasi-homogeneous foliation of admissible type, as we have imposed to the definition the condition of being a generalized surface. In general, such a foliation may have more separatrices, or be dicritical. In \cite[Thm. 5.2]{FMNeciosup2} a sufficient condition is stated for being a generalized surface. More precisely, if we write $G(\Psi,z)=\displaystyleplaystyle\sum_{\alpha,\begin{enumerate}ta}G_{\alpha,\begin{enumerate}ta}\Psi^{\alpha}z^{\begin{enumerate}ta}$, and
$$\nu_{2,r}(G):=\min\left\{{2\alpha+r\begin{enumerate}ta\over \gcd(2,r)}; G_{\alpha,\begin{enumerate}ta}\neq0\right\},$$
then, if $\nu_{2,r}(G)\geq{r-2\over \gcd(2,r)}$, the foliation is a generalized surface.
On the other hand, let us observe that if $d_i=1$ for all $i$, $\varphi$ is reduced and the singular locus of ${\mathcal F}_{\Omegaega}$ is the origin of coordinates. By Frobenius singular Theorem \cite{Malgrange}, ${\mathcal F}_{\Omegaega}$ admits holomorphic first integral and the study of these foliations is then equivalent to that of the surfaces. We will suppose, at any time, that $d_i>1$ for some $i$, and we will denote $\Sigma_{p,q}^{(d_1,\cdots,d_l)}$ the set of integrable 1-forms analytically equivalent to a 1-form as in (\ref{forma_quasi_homogeneous}), with $d_i>1$ for some $i$.
\section{Desingularization of quasi homogeneous foliations}\label{section3}\label{Desingularization}
Let us consider a quasi-homogeneous foliation of admissible type, ${\mathcal F}_{\Omegaega}$, generated by an integrable 1-form $\Omegaega\in\displaystyleplaystyle \Sigma_{p,q}^{d_1,\cdots,d_l}$, according with the notations of Section \ref{general}. The reduction of the singularities of ${\mathcal F}_{\Omegaega}$ is achieved after the reduction of its separatrices. We use in this paper a precise reduction, namely the one obtained by Weierstrass-Jung method, that we shall sketch here for the sake of completeness. This precise reduction will be useful in the sequel in order to extend the conjugation of the holonomy to a whole neighbourhood of the exceptional divisor.
This reduction will be described in three steps:
{\bf Step I}: The separatrix $S$ being defined by $z^2+\varphi(x,y)=z^2+\displaystyleplaystyle\prod_{i=1}^{l}(y^p-a_ix^q)^{d_i}=0$, we shall proceed to desingularize first the curve $\varphi(x,y)=0$. This is done algorithmically taking into account its characteristic exponents, more precisely, the continuous fraction expansion of $\displaystyleplaystyle{p\over q}=[c_0;c_1,\cdots,c_N]$. It is necessary to do $k=\displaystyleplaystyle\sum_{\nu
=0}^{N}c_{\nu}$ quadratic transformations (point blow-ups), whose composition will be denoted $\pi_I:$
$$\pi_I:(M_I,E_I)\to({\mathbb C}^3,{\bf 0}).$$
Denote $\widetilde{\Omegaega}$ the strict transform of $\Omegaega$. Locally, in appropriate coordinates, it is given as
$$\widetilde{\Omegaega}=\displaystyleplaystyle(z^2+x^ay^bU_{\alpha})\omegaega_{\alpha}+xy\epsilonta_{\alpha},$$ where $a,b$ are natural numbers, depending on $p,q$ and the chosen chart, $\omegaega_{\alpha}=mydx+nxdy$ is a linear form, with $m$, $n\in {\mathbb N}$, $U_{\alpha}$ is a holomorphic function, that in the ``most interesting'' chart obtained after the last blow-up is
$\displaystyleplaystyle U_{\alpha}=h^r=\left(\prod_{i=1}^l(y^{\delta}-a_i)^{d_i'}\right)^r$, where $r=\gcd(d_1,\cdots,d_l)$, $d_i'=d_i/r$, $\delta=\gcd (p,q)$ and $\displaystyleplaystyle\epsilonta_{\alpha}=d\left(z^2+x^ay^bU_{\alpha}\right)+\Delta_{\alpha}\cdot \left(a{dx\over x}+b{dy\over y}+{dU_{\alpha}\over U_{\alpha}}-2{dz\over z}\right)$. Here $\Delta_{\alpha}$ is a certain germ of holomorphic function that we will not precise here.
{\bf Step II:} Consider the foliation defined by $\widetilde{\Omegaega}$. We will blow-up certain curves biholomorphic to $\mathbb{P}_{{\mathbb C}}^1$. This process will depend of the nature of integers $a$, $b$, and follows the scheme of the quasi-ordinary case studied in \cite{FM}. More precisely:
{\bf II.a) If $d$ is even,} $a, b$ are also even. Blow-up $(z=y=0)\:\; {a\over 2}$ times and $(z=x=0)\:\; {b\over 2}$ times. The final result is schematized in Figure \ref{figura2}.
\begin{enumerate}gin{center}
\includegraphics[width=11.5cm]{Caso-i-final-segunda-etapa}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{_{II}}$ of $S$. }\label{figura2}
\epsilonnd{center}
{\bf II.b) If $d$ is odd, $p$ even and $q$ odd}, a certain number of suitable monoidal transformations, depending on the chart, is necessary to perform. The final result is schematized in Figure \ref{figuraCasoii_1}.
\begin{enumerate}gin{center}
\includegraphics[width=10.5cm]{Caso-ii1-final-segunda-etapa}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{II}$ of $S$ (Case II.b)}\label{figuraCasoii_1}
\epsilonnd{center}
{\bf II.c) If $d, a$ and $b$ are odd}, after the sequence of monoidal transformations, a final quadratic transformation will be necessary to do, as in the quasi-ordinary case of \cite{FM}. Figure \ref{figure_trans_monoidal_caso_impar} represents the final result in an appropriate chart.
\begin{enumerate}gin{center}
\includegraphics[width=11cm,height=3.5cm]{impar-ultima-componente}\\
\figcaption{$\widetilde{S}$, strict transform by $\pi_{II}$ of $S$ (Case II.c)}\label{figure_trans_monoidal_caso_impar}
\epsilonnd{center}
We shall denote $\pi_{II}$ the composition of the transformations done in this step.
{\bf Step III}. At the end of Step II, there exist analytic coordinates in which the local equation of the strict transform of the foliation is
$$\Omegaega_{PQ}=(t^2+h^r)\omegaega_{PQ}+xy\epsilonta_{PQ},$$
where
$$\begin{enumerate}gin{array}{rcl}
\omegaega_{_{PQ}} & = & ({pq\over\delta}d)ydx+(nqd)xdy\\
\epsilonta_{_{PQ}} & = & d(t^2+h^r)+x^{{pq\over \delta}d'(1-{r\over2})}y^{nqd'(1-{r\over 2})}th.G_{1PQ}\Bigg(r{dh\over h}-2{dt\over t}\Bigg)\\
G_{1PQ} & = & G\Big(x^{{pq\over\delta}d'}y^{nqd'}h,x^{{pq\over
\delta}{d\over2}}y^{nq{d\over2}}t\Big).\\
P & = & {pq\over\delta}d-2\left({p+q\over\delta}-1\right).\\
Q & = & nqd-(m+n-1).
\epsilonnd{array}$$
It is necessary to blow-up the lines $z=0, y=a_i^{1\over \delta}$, according to the nature of each $d_i$. The components of the divisor are, topologically, $\mathcal{A}_j\times \mathbb{D}$, where $\mathcal{A}_j\approx\mathbb{P}_{{\mathbb C}}^1$ and $\mathbb{D}$ a disc around the origin.
Denote $\widetilde{D}$ the last component generate in Step II, i.e., the main component where the separatrix cuts the exceptional divisor. This component will be called {\it essential component}.
Note that the modifications done in this step do not alter the topology of this component. This will be precised in Section \ref{section_toplogy_of_the_divisor}.
Analogously as in the previous steps, $\pi_{III}$ will denote the composition of the transformations done here.
\section{Topology of the divisor}\label{section_toplogy_of_the_divisor}
Let $\widetilde{{\mathcal F}}$ be the strict transform of a quasi-homogeneous holomorphic foliation of admissible type via the morphism $\pi:(M,E)\to({\mathbb C}^3,{\bf0})$ ($\pi=\pi_{_{I}}\circ\pi_{_{II}}\circ\pi_{_{III}}$), as described in Section \ref{section3}.
With the conditions imposed, the singular locus $\mathcal{S}:=\Sing(\widetilde{{\mathcal F}})$, is an analytic space of dimension $1$ with normal crossings. $\mathcal{S}$ is given by the intersection of the components of the divisor $E$, along with the intersection of the strict transform of $S$ with the divisor. These components are denoted as $D_{\alpha},\; D_{\alpha j},\; \mathcal{A}_j\times \mathbb{D}$ (see Section \ref{section3} for notations).
It will be useful, in order to study the projective holonomy of the foliation, to know their homotopy type, once we have removed the singular locus. Let us describe it briefly in this section.
In the first step of the reduction we have produced several components $D_{\alpha}$. After removing the singular locus they are homotopically equivalent to:
$$D_{\alpha}\smallsetminus\mathcal{S}\approx
\left\{
\begin{enumerate}gin{array}{ll}
{\mathbb C}\times{\mathbb C}, & \hbox{if}\; \alpha=1; \\\\
{\mathbb C}\times{\mathbb C}^*, & \hbox{if}\;1<\alpha\leq c_0+1; \\\\
{\mathbb C}^*\times{\mathbb C}^*, & \hbox{ in other cases.}
\epsilonnd{array}
\right.
$$
The components $D_{\alpha j}$ appearing in the second step, removing the singular locus, are homotopically equivalent to ${\mathbb C}\times{\mathbb C}^*$, to ${\mathbb C}^*\times{\mathbb C}^*,$ to ${\mathbb C}\times({\mathbb C}\smallsetminus\{\text{2 points}\})$, or, in the most interesting case, to $({\mathbb C}^*\times{\mathbb C})\smallsetminus\mathcal{C}$, where, in coordinates $(y,t), \mathcal{C}$ is the affine curve defined by $t^2-y^{a}.v=0$, where $v$ is a unit, or
$$t^2-\displaystyleplaystyle\left(\prod_{i=1}^{l}(y^{\delta}-a_i)^{d'_i}\right)^{r}=0,$$
with previous notations.
Finally, Step III produces new components homotopically equivalent to ${\mathbb C}\times{\mathbb C}$, ${\mathbb C}^*\times{\mathbb C}$ or $({\mathbb C}\smallsetminus\{\text{2 points}\} )\times{\mathbb C}$, that are not relevant for our study. However, we highlight the component $D_{\alpha j}$ (generated in the second step) which is modified now to $\widetilde{D}$ (essential component defined in Section \ref{Desingularization}), but whose topology does not change in this step and consequently $\widetilde{D}\smallsetminus\Sing(\widetilde{{\mathcal F}}_{\Omegaega})\thickapprox ({\mathbb C}^*\times{\mathbb C} )\smallsetminus\mathcal{C}$.
The fundamental group of $({\mathbb C}^*\times{\mathbb C})\smallsetminus\mathcal{C}$ can be computed using the Zariski-Van Kampen method, as described, for instance, in \cite{ACT} (see also \cite{VK} as a classical reference). Even if it is standard enough, for the sake of completeness we shall briefly review here this method. Consider the projection $\rho:{\mathbb C}^2\to{\mathbb C},\;\rho(x,t)=x$.
Let us denote by $\Delta:=\{a_1,a_2,\cdots,a_l\}$ and $\mathcal{L}:=\displaystyleplaystyle\begin{itemize}gcup_j\rho^{-1}(a_j)$. The restriction \begin{enumerate}gin{equation}\label{fibracion_localmente_trivial}
\rho:{\mathbb C}^2\smallsetminus\mathcal{C}\cup\mathcal{L}\to{\mathbb C}\smallsetminus\Delta,
\epsilonnd{equation}
is a locally trivial fibration, with fibres isomorphic to ${\mathbb C}\smallsetminus\{\text{2 points}\}$. So, for every point in ${\mathbb C}\smallsetminus\Delta$, we obtain the exact sequence
$$
\xymatrix{1\ar[r] & \pi_1\left({\mathbb C}\smallsetminus\{\text{2 points}\}\right)\ar[r]^-{i_{*}} & \pi_1({\mathbb C}^2\smallsetminus \mathcal{C}\cup\mathcal{L}) \ar[r]^-{\rho_{*}}& \ar@/_2pc/[l]^-{s_{*}}\pi_1({\mathbb C}\smallsetminus\Delta)\ar[r]& 1},
$$
where $i^*$ and $s^*$ are map induced in homotopy from inclusion and section of $\rho$ respectively.
Let $\mathbb{D}\subset {\mathbb C}$ be a sufficiently big closed disk, such that $\Delta$ is contained in its interior. Choose a point $\star$ on $\partial \mathbb{D}$. Take a small disk $\mathbb{D}_j$ centered at ${a_j}\in\Delta$ containing no other elements of $\Delta$ and choose a point $R\in\partial\mathbb{D}_j$. Consider a path $\alpha\in{\mathbb C}\smallsetminus\Delta$ joining $R$ and $\star$, and denote by $\epsilonta_{R,\mathbb{D}_j}$ the closed path based at $R$ that runs counterclockwise along $\partial\mathbb{D}_j$. The homotopy class of the loop $\gamma_j:=\alpha^{-1}.\epsilonta_{R,\mathbb{D}_j}.\alpha$ is called a \textit{meridian} of $a_j$ in ${\mathbb C}\smallsetminus\Delta$. Then, the collections of meridians in ${\mathbb C}\smallsetminus\Delta$ (one for each point of $\Delta$) define bases of $\pi_1({\mathbb C}\smallsetminus\Delta,\star).$ Similarly we can choose meridians $g_1,g_2$ in a fiber of the restriction (\ref{fibracion_localmente_trivial}) and $\gamma$ a meridian around the straight line $x=0$. It follows that
$$
\pi_1( ({\mathbb C}^*\times {\mathbb C}) \smallsetminus \mathcal{C})=\Big\langle g_1,g_2,\gamma;\ g_i^{\sigma^r}=g_i\;\; \wedge \;\; g_i^{\sigma^b}=\gamma^{-1}g_i\gamma\Big\rangle,
$$
where $g_i^{\sigma^r}$ is the action on $g_i$ of $\gamma_j$ (known as the factorization of the braid monodromy, see \cite{ACT}).
The expression of the group can be simplified obtaining:
$$\pi_1(({\mathbb C}^*\times {\mathbb C} ) \smallsetminus \mathcal{C})=
\left\{
\begin{enumerate}gin{array}{ll}
\Big\langle \alpha, \begin{enumerate}ta,\gamma: \begin{enumerate}ta\alpha^r=\alpha^r\begin{enumerate}ta\; \wedge\; \gamma\alpha=\alpha\gamma \Big\rangle, & \hbox{if $r=2m+1$ is odd;} \\\\
\Big\langle \alpha, \begin{enumerate}ta,\gamma: \alpha^r=\begin{enumerate}ta^2\; \wedge\; \gamma\alpha=\alpha\gamma \Big\rangle, & \hbox{if $r=2m$ is even.}
\epsilonnd{array}
\right.
$$
where $\alpha:=g_2g_1$ and $\begin{enumerate}ta:=(g_2g_1)^mg_2$.
\section{Holonomy of the essential component}\label{section_holonomy_of_the_essential_component}
The dynamical behavior of one foliation may be studied in a neighbourhood of a leaf by the representation of its fundamental group, as introduced by C. Ehresmann in 1950. In this study we will mainly follow J.-F. Mattei and R. Moussu \cite{Mattei-Moussu}: let us choose a point ${\bf p}$ of a leaf $L$ and a germ of a transversal section $\Sigma$ in ${\bf p}$. The lifting of a closed path $\gamma$ starting in ${\bf p}$, following the leaves of the foliation, induces germs of diffeomorphisms
$$h_{\gamma}:(\Sigma,{\bf p})\to (\Sigma,{\bf p}),$$
that only depend on the homotopy class of the path. The map $h_{\gamma}$ is called {\it holonomy } of the leaf $L$. The representation of the holonomy of $\pi_1(L,{\bf p})$ is the morphism defined by
$$\begin{enumerate}gin{array}{cccc}
Hol(L)&:\pi_1(L,{\bf p})&\to & Diff(\Sigma,{\bf p})\\
& \gamma &\longmapsto & h_{\gamma},
\epsilonnd{array}$$
and the {\it holonomy group} of the foliation along $L$ is the image of this application $Hol(L)$. Different points in the leaf and different transversal sections define conjugated representations.
In order to lift the path $\gamma$ it is necessary to have a fibration, that is transverse to the foliation. Let us describe it. In general let $\Omegaega$ be an integrable 1-form in $({\mathbb C}^3,{\bf 0})$, $\pi:(M,E)\to ({\mathbb C}^3,{\bf 0})$ a minimal reduction of singularities of the foliation ${\mathcal F}_{\Omegaega}$. Let $\widetilde{{\mathcal F}}_{\Omegaega}$ be the strict transform of ${\mathcal F}_{\Omegaega}$ under $\pi$ and $D$ a component of the exceptional divisor $E$.
\begin{enumerate}gin{defin}\label{Def_Hopf_fibration}
A {\bf Hopf fibration adapted to ${\mathcal F}_{\Omegaega}$}, $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$, is a holomorphic fibration $f:M\to D$ transverse to the foliation ${\mathcal F}_{\Omegaega}$, i.e.,
\begin{enumerate}gin{enumerate}
\item $f$ is a submersion and $f|_{D}=Id_{D}.$
\item The fibres $f^{-1}(p)$ of $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$ are contained in the separatrices of $\widetilde{{\mathcal F}}_{\Omegaega}$, for all $p\in D\cap\Sing(\widetilde{{\mathcal F}}_{\Omegaega})$.
\item The fibres $f^{-1}(p)$ of $\mathcal{H}_{{\mathcal F}_{\Omegaega}}$ are transverse to the foliation ${\mathcal F}_{\Omegaega}$, for all $p\in D\smallsetminus\Sing(\widetilde{{\mathcal F}}_{\Omegaega})$.
\epsilonnd{enumerate}
\epsilonnd{defin}
Consider now the particular case $\Omegaega\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$. The vector field
$$\mathcal{X}=px{\partial\over\partial x}+qy{\partial\over \partial y} + {pqd\over 2}z{\partial\over\partial z}. $$
verifies $\Omegaega(X)=pqd(z^2+\varphi)$, so, it is transverse to ${\mathcal F}_{\Omegaega}$ outside the separatrix $S$, and $S$ is a union of trajectories of $\mathcal{X}$ (equivalently, $S$ is invariant by $\mathcal{X}$). The trajectories $\mathcal{X}$ give the fibres of the Hopf fibration adapted to $\Omegaega$.
Let $\widetilde{D}$ be the essential component of the reduction of singularities of ${\mathcal F}_{\Omegaega}$, as defined in Section \ref{Desingularization}.
\begin{enumerate}gin{defin}
The \textbf{exceptional holonomy group} is the group of holonomy of $\widetilde{D}$. Let us denote it $H_{\Omegaega,\widetilde{D}}$.
\epsilonnd{defin}
As a consequence of the results of Section \ref{section_toplogy_of_the_divisor}, $H_{\Omegaega,\widetilde{D}}$, is generated by elements $h_{\alpha}, h_{\begin{enumerate}ta}, h_{\gamma}$, which are the holonomy diffeomorphisms of, respectively, paths $\alpha, \begin{enumerate}ta, \gamma$.
Consider, now, two such foliations ${\mathcal F}_{\Omegaega_1}, {\mathcal F}_{\Omegaega_2}$, where $\Omegaega_1,\Omegaega_2\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$, with respective exceptional holonomy groups
$$H_{\Omegaega_1,\widetilde{D}}=\langle h_{\alpha}^{1},h_{\begin{enumerate}ta}^{1},h_{\gamma}^{1}\rangle, H_{\Omegaega_2,\widetilde{D}}=\langle h_{\alpha}^{2},h_{\begin{enumerate}ta}^{2},h_{\gamma}^{2}\rangle$$
represented on a the transverse section. Suppose they are analytically conjugated by an element $\psi\in Diff({\mathbb C},0)$ such that
$$\psi^*(h_{\alpha_i}^1):=\psi^{-1}h_{\alpha_i}^1\psi=h^2_{\alpha_i};\; \alpha_i\in\{\alpha,\begin{enumerate}ta, \gamma\}.$$
Then we have the following result.
\begin{enumerate}gin{lema}\label{lema-extension-de-holonomia--}
There exist a fibered diffeomorphism $\phi, \phi(x,{\bf p})=(\varphi(x,{\bf p}),{\bf p})$ between two open neighbourhoods $V_j$ of $\widetilde{D}$ in the space $(M,E)$ such that
\begin{enumerate}gin{enumerate}
\item $\phi$ sends the leaves of the foliation ${\widetilde{{\mathcal F}}_{\Omegaega_1}}{|_{V_1}}$ to the leaves of the foliation ${\widetilde{{\mathcal F}}_{\Omegaega_2}}{|_{V_2}}$.
\item The restriction of $\phi$ to the transverse section $\Sigma$ is $\psi$.
\epsilonnd{enumerate}
\epsilonnd{lema}
\begin{enumerate}gin{proof}
Let ${\bf p}\in \mathcal{L}=\widetilde{D}\smallsetminus \mathcal{S}$ and $\gamma_{\bf p}\subset \mathcal{L}$ be a path from ${\bf p}$ to ${\bf p}_j$. We have that the foliation ${\mathcal F}_{\Omegaega_j}$ is generically transverse to the Hopf fibration relative to $\widetilde{D}$, $\mathcal{H}_{{\mathcal F}_{\Omegaega_j}}$, outside the strict transform of the separatrix $\widetilde{S}$ and of the components $D_{\alpha j}, \mathcal{A}_j$, such that $\widetilde{D}\cap D_{\alpha j}\neq\epsilonmptyset$, $\widetilde{D}\cap \mathcal{A}_j\neq\epsilonmptyset$.
The projection $({\mathbb C},0)\times\mathcal{L}\to \mathcal{L}$, is locally a covering map. Then, for each point $(x,{\bf p})\in{\mathbb C}\times\mathcal{L}$, with $|x|$ small enough, we can consider the lifting $\widetilde{\gamma}^1_{\bf p}$, following a leaf of the foliation $\widetilde{{\mathcal F}}_{\Omegaega_1}$, of the path $\gamma_{\bf p}$, such that $\widetilde{\gamma}_{\bf p}^1(0)=(x,{\bf p}).$\\
If $(x_1,{\bf p}_1)=\widetilde{\gamma}^1_{\bf p}(1)$ is the end point of $\widetilde{\gamma}^1_{\bf p}$, we lift the path $\gamma^{-1}_{\bf p}$ to $\widetilde{\gamma}^2_{\bf p}$ in the foliation $\widetilde{{\mathcal F}}_{\Omegaega_2}$ such that $\widetilde{\gamma}^{2}_{\bf p}(0)=(\psi(x_1),{\bf p}_2)$.
Let $(x_2,{\bf p})=\widetilde{\gamma}^{2}_{\bf p}(1)$. Define
$$\phi(x,{\bf p})=(x_2,{\bf p})$$
as $\phi_{|\Sigma_1}=\psi,\; \phi(x,{\bf p})$ does not depend on the chosen path.
Let us observe that we can define $\phi$ as close as we want to the points ${\bf q}\in\mathcal{S}$, in $\widetilde{D}$. In fact, $\widetilde{D}\smallsetminus \mathcal{S}\cong {\mathbb C}^*\times{\mathbb C}\smallsetminus \mathcal{C}$, so ${\bf q}\in \mathcal{C}$ or ${\bf q}\in{\mathbb C}^*$. We can consider
a meridian relative to ${\bf q}$ with base point ${\bf p}$, (see Section \ref{section_toplogy_of_the_divisor}), and define the path $\begin{enumerate}ta_{\bf p}=\alpha_{\bf q}\gamma_{\bf p}$ with starting point ${\bf p}$ and end point ${\bf p}_j$. The path $\gamma^{-1}_{\bf p}\begin{enumerate}ta_{\bf p}$ is a meridian relative to ${\bf q}$ with base point ${\bf p}_j$, so its homotopy class define an element of $\pi_1(\mathcal{L},{\bf p}_j)$
and
$$h^1_{\gamma_{\bf p}^{-1}\begin{enumerate}ta_{\bf p}}=\psi^*(\gamma_{\bf p}^{-1}\begin{enumerate}ta_{\bf p}).$$
Therefore $\phi$ extends to a neighbourhood of $\widetilde{D}\smallsetminus \mathcal{S}$.
On the other hand, the singular points ${\bf q}\in\mathcal{S}$ are points of intersection of $\widetilde{D}$ with the other components of the divisor, and with the separatrix. These points are of dimensional type two or three and in a neighbourhood of all them, $\widetilde{D}$ is a strong separatrix\footnote{In dimension two, \textit{strong} separatrices around simple singular points are the ones corresponding to non-zero eigenvalues of the linear part. In higher dimension, the separatrix is called \textit{strong} if it corresponds to a strong separatrix in dimension two for a generic plane transversal section.}
From \cite{Cerveau-Mozo}, it follows that the conjugation of the holonomy of $\widetilde{D}$ implies the conjugation of the reduced foliations in a neighbourhood of these points. This conjugation coincides, outside of the separatrix, with the diffeomorphism $\phi$, and in consequence, $\phi$ is extended to a diffeomorphism around $\widetilde{D}$.
\epsilonnd{proof}
\section{Analytic classification}\label{section_cladification}
In this section we will study the analytic classification of quasi-homogeneous cuspidal foliations ${\mathcal F}_{\Omegaega}$ in $({\mathbb C}^3,{\bf 0})$, using as the main tool the lifting of the projective holonomy, as stated in previous sections.
As described in Section \ref{section_toplogy_of_the_divisor}, after the reduction of singularities of a quasi-homogeneous cuspidal holomorphic foliation in $({\mathbb C}^3,{\bf 0})$, the first component, $D_1$, of the exceptional divisor $E$, is either
\begin{enumerate}gin{itemize}
\item $D_1\smallsetminus \mathcal{S}\simeq {\mathbb C}^*\times{\mathbb C}^*$, if the separatrix of ${\mathcal F}_{\Omegaega}$ is defined by the equation
$$z^2+x^{n_1}y^{n_2}\prod_{i=1}^{l}\begin{itemize}g(y^{p}-a_ix^{q}\begin{itemize}g)^{d_i}=0,\ \text{with } (n_1,n_2)\neq (0,0),$$
\item or $D_1\smallsetminus \mathcal{S}\simeq {\mathbb C}^2$, if the separatrix of ${\mathcal F}_{\Omegaega}$ is defined by the equation
$$z^2+\prod_{i=1}^{l}\begin{itemize}g(y^{p}-a_ix^{q}\begin{itemize}g)^{d_i}=0.$$
\epsilonnd{itemize}
In the admissible case that we are studying, $D_1\smallsetminus \mathcal{S}$ is simply connected, so, the existence of a first integral around it is guaranteed due to the results of Mattei and Moussu \cite{Mattei-Moussu}. In order to be able to extend this first integral, and consequently, to extend the conjugation diffeomorphism, it would be necessary to impose additional technical conditions on one, or possibly several components of the exceptional divisor, that we will be call in the sequel \textit{special components}. These special components will be
those that arise at the end of the monoidal transformations with center the projective lines $D_1\cap S_1$, $D_{c_0+1}\cap S_{c_0+1}$ that we will denote by $\widetilde{D}', \;\widetilde{D}''$ respectively. Note that
$$\widetilde{D}'\smallsetminus \mathcal{S}\simeq{\mathbb C}\times({\mathbb C}\smallsetminus\{2pts\})\approx \widetilde{D}''\smallsetminus \mathcal{S}.$$
This will motivate the following definition:
\begin{enumerate}gin{defin}\label{defini_propiedades_p_1andp_2}
Let $\Omegaega\in\Sigma_{p,q}^{{(d_1,\cdots,d_l)}}$ be, we will say that the foliation ${\mathcal F}_{\Omegaega}$:
\begin{enumerate}gin{enumerate}
\item For $d-$even ({\bf Case i}), satisfies the property $\wp_1$: if the holonomy of the leaves $\widetilde{D}'\smallsetminus \mathcal{S},\; \widetilde{D}''\smallsetminus \mathcal{S}$, is linearizable.
\item For $d-$odd ({\bf Case ii.1}), satisfies the property $\wp_2$: if the holonomy of the leaf $\widetilde{D}''\smallsetminus \mathcal{S}$, is linearizable.
\epsilonnd{enumerate}
\epsilonnd{defin}
The following theorem is the main result of this paper.
\begin{enumerate}gin{teorema}\label{thm_clasificacion_analitica}
Let $\Omegaega_1, \Omegaega_2$ be elements of $\Sigma_{p,q}^{(d_1,\cdots,d_l)}$. Consider the foliations ${\mathcal F}_{\Omegaega_1}$ and ${\mathcal F}_{\Omegaega_2}$ that satisfy one of the properties $\wp_1, \wp_2$ if we are in one of the cases described above, with exceptional holonomy groups $H_{\Omegaega_i, \tilde{D}}=\langle h_{g_1}^i,h_{g_2}^i,h_{\alpha}^i\rangle, \; i=1,2$. Then, the foliations are analytically conjugated if and only if the triples $(h_{g_1}^i,h_{g_2}^i,h_{\alpha}^i)$ are also analytically conjugated.
\epsilonnd{teorema}
\begin{enumerate}gin{proof}
If the foliations are conjugated, then their essential holonomy groups are conjugated. The arguments are exactly the same that those described in \cite{Cerveau-Mozo}.
Assume that the holonomies are conjugated via $\Psi$, and let $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ be the respective strict transform of the foliations ${\mathcal F}_{\Omegaega_i}$, after its reduction of singularities. From the existence of the Hopf fibration relative to $\widetilde{D}$ (see Figure \ref{secuencia-explosiones-segun-di}) and Lemma \ref{lema-extension-de-holonomia--}, $\Psi$ may be extended to a neighbourhood $V_i\subset (M,E)$ of $\widetilde{D}$ and as a consequence, we have that $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ are conjugated in $V_i$. Now, we need to conjugate the foliations in a neighbourhood of all the other components of the divisor. We must first check the existence of the first integral around of the singular points outside $\widetilde{D}$. In fact, note that $D_1\smallsetminus \mathcal{S}$ is simply connected, so its holonomy group is trivial. As a consequence, the holonomy of the leaf $D_{2}\smallsetminus \mathcal{S}\approx {\mathbb C}\times{\mathbb C}^{*}$, generated by a loop around $L_1:=D_1\cap D_{2}$, is periodic (see \cite{Mattei-Moussu}). The same argument proves that $D_{\alpha}\smallsetminus \mathcal{S}$ has periodic holonomy, for all $1<\alpha\leq c_0+1$. So, $\widetilde{{\mathcal F}}_{{\Omegaega}_i}$ has first integral around
$$L_{\alpha}:=D_{\alpha}\cap D_{\alpha+1},\;1\leq\alpha \leq c_0+1.$$
On the other hand, the leaves $D_{\alpha}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}^*\times{\mathbb C}^*$ have periodic holonomy, generated by two loops around the lines
$$L_{\alpha}=D_{\alpha}\cap D_{\alpha+1};$$
$$L_{s_\nu}^{\alpha}=D_{\alpha}\cap D_{s_\nu}.$$
The above arguments prove the existence of the first integral around the lines
$$L_{\alpha j}:=D_{\alpha j}\cap D_{\alpha(j-1)}$$
where $\alpha, j$ are such that $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox{\mathbb C}\times{\mathbb C}^*$ or $\thickapprox {\mathbb C}^*\times{\mathbb C}^*$.
Finally, we need to guarantee the existence of the first integral around the points that represent the intersection of the divisor with the strict transform of the separatrix. These points are in the components $D_{\alpha j}$, and:
\begin{enumerate}gin{itemize}
\item Either $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$,
\item or $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}^*\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$,
\item or $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox ({\mathbb C}^*\times{\mathbb C} )\smallsetminus \mathcal{C}$.
\epsilonnd{itemize}
Let us denote by $\mathcal{D}:=D_{\alpha j}$ such that $D_{\alpha j}\smallsetminus \mathcal{S}\thickapprox {\mathbb C}\times({\mathbb C}\smallsetminus\{{\rm 2\;points}\})$.
Note that
$\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$ or $\mathcal{D}\cap\widetilde{S}=\mathcal{L}$, where $\widetilde{S}$ represents the strict transform of the separatrix, and $\mathcal{L},\mathcal{L}_1,\mathcal{L}_2$ are projective lines.
Let us suppose now that $d$ is even, then there exist exactly two components of the exceptional divisor such that $\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$. The points of $\mathcal{L}_1, \mathcal{L}_2$ are singular points of dimensional type two.
Assuming that $\Omegaega_j\in\Sigma_{p,q}^{(d_1,\cdots,d_l)}$, ${\mathcal F}_{\Omegaega_j}$ satisfies the property $\wp_1$, i.e., the holonomy of the leaves $\mathcal{D}\smallsetminus \mathcal{S}$ is linearizable. This fact guarantees the existence of the first integral around $\mathcal{L}_1,\; \mathcal{L}_2$.
\begin{enumerate}gin{center}
\includegraphics[width=10cm]{TorreSuperfiesCaso-Par-existencia-integral-primera}
\figcaption{Existence of the first integral around the points $\rho_i$ and $\mu_i$}\label{sin nombre}
\epsilonnd{center}
Now let us prove the existence of the first integral around the points (see Figure \ref{sin nombre}) $$\rho_i:=D_{\alpha j}\cap\widetilde{S}\cap\mathcal{D}.$$
Let $h_{\alpha_i}$ be the holonomy associated to $\alpha_i$, loop around $\mathcal{L}_i$; $h_{\alpha_i}$ is linearizable. Now let us consider $\begin{enumerate}ta\subset\mathcal{D}\smallsetminus\mathcal{S}\simeq {\mathbb C}\times({\mathbb C}\smallsetminus\{2\;{\rm points}\})$, a loop around $D_{\alpha j}\cap\mathcal{D}$. Note that $\begin{enumerate}ta\subset {\mathbb C}\times\{1\;{\rm point}\}\simeq{\mathbb C}$; so, $\begin{enumerate}ta$ is homotopically trivial and the associated holonomy to $\begin{enumerate}ta$ is $1_{{\mathbb C}}$, the identity map, $h_{\begin{enumerate}ta}=1_{{\mathbb C}}$. Then, the holonomy group of $D_{\alpha j}\smallsetminus \mathcal{S}$ is linearizable and therefore, around $\rho_i$, $\widetilde{\Omegaega}_i$ is linearizable. So, there exists a first integral around $\rho_i$. Consider now the points $\mu_i$ (see Figure \ref{sin nombre}), and $h_\gamma$, the holonomy associated to $\gamma$, loop around the projective line $D_{(\alpha+1)j}\cap D_{\alpha j}$. Note that $\gamma$ is such that $\gamma^{-1}$ is a loop around $D_{\alpha-1}\cap D_{\alpha j}$, so $h_{\gamma^{-1}}$ is the holonomy of the leaf $D_{\alpha-1}\smallsetminus \mathcal{S}$, which is periodic. Then around $\mu_i$ there exists a first integral. In a similar way, a first integral can be found around $E\cap \widetilde{S}$.
Let us suppose that $d,q$ are odd and $p$ is even. There exist exactly a component $\mathcal{D}$ for which $\mathcal{D}\cap\widetilde{S}=\mathcal{L}_1\cup \mathcal{L}_2$ and a finite number of components for which we have $\mathcal{D}\cap\widetilde{S}=\mathcal{L}$. It is easy to see that around $\mathcal{L}$ there exist a first integral, and the existence of first integral around the lines $\mathcal{L}_1, \mathcal{L}_2$ follows, from property $\wp_2$, as in the previous case.
Let us see now the extension of $\Psi$ around all the exceptional divisor. The idea is, first, to extend $\Psi$ to a neighborhood of all the components of the divisor in which the separatrix intersects, and finally to extend it to the rest of the components. In both situations we should respect the fibration and the first integral that we have constructed previously. We have that $\widetilde{{\mathcal F}}_{{\Omegaega}_j}$ are analytically conjugated, via $\Psi$, in a neighbourhood of $\widetilde{D}$. We want to extend $\Psi$ to a neighbourhood of the exceptional divisor.
$$E=E_I\cup E_{II}\cup E_{III}.$$
Note that $E_{III}=\begin{itemize}gcup\mathcal{A}_j\times{\mathbb C}$, and these components are topologically equivalent to ${\mathbb P}_{{\mathbb C}}^1\times\mathbb{D}$. As $\widetilde{{\mathcal F}}_{{\Omegaega}_j}$ are analytically conjugate in a neighbourhood of $\widetilde{D}$, it follows that $\Psi$ is extended to a neighbourhood of $\widetilde{D}\cup E_{III}$
\begin{enumerate}gin{figure}[h]
\begin{enumerate}gin{center}
\includegraphics[width=11.5cm]{extension-analitica-1}\\
\figcaption{Extension of $\Psi=\Psi_2\Theta\Psi_1^{-1}$ to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$.}\label{extension-analitica}
\epsilonnd{center}
\epsilonnd{figure}
On the other hand, for the case that $d$ is even, around ${\bf p}:=D_{\alpha-2}\cap D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$, with $\alpha=s_{_N}$ and $\begin{enumerate}ta={\tilde{Q}_2\over2}$ (where $\tilde{Q}= \left( \frac{pq}{\delta} -nq\right) d-2\left( \frac{p+q}{\delta} -(m+n+1)\right)$, see Figure \ref{extension-analitica}) a first integral is given by the equation
$$F_j:= x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}U_j(x,s,t);\ U_j({\bf 0})\neq0$$
where for simplicity we denote $x:=x_{\alpha-1};\; \varepsilonilon=\begin{enumerate}ta-1,\; \nu=N\;; j_{\nu}=c_{\nu}-1$.
Let $$\mathcal{B}_c=\Big\{x\in{\mathbb C}: |x|<c\Big\}$$ be, for $c\in\mathbb{R}^{+}$ large enough, and
$$D_{\varepsilon}:=\{s\in{\mathbb C}:|s|<\varepsilon_1\}\times\{t\in {\mathbb C}: |t|<\varepsilon_2\}.$$
Similar arguments as in \cite{Meziani} and \cite{FM} are now used. First define a diffeomorphism $\Psi_j=(\Psi_{j1},\Psi_{j2},\Psi_{j3})$, on an open set $\mathcal{B}_c\times D_{\varepsilon}$, that transforms the first integral in $x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}$ and respects the fibration, i.e.:
\begin{enumerate}gin{equation}\label{trasforma-Int.Primera}
\Psi_{j1}^{m_{\nu}}\Psi_{j2}^{n_{\nu}+2(\varepsilonilon-1)}\Psi_{j3}^{n_{\nu}-2\varepsilonilon}=x^{m_{\nu}}s^{n_{\nu}+2(\varepsilonilon-1)}t^{n_{\nu}-2\varepsilonilon}U_j,
\epsilonnd{equation}
and
\begin{enumerate}gin{equation}\label{respeta-fibracion}
\begin{enumerate}gin{array}{ll}
\Psi_{j3}\Psi_{j1}^{{P\over2}-\widetilde{Q}_2+2}& =tx^{{P\over2}-\widetilde{Q}_2+2}\\\\
\Psi_{j2}\Psi_{j1}^{\widetilde{Q}_2-{P\over2}} & =sx^{\widetilde{Q}_2-{P\over2}}.
\epsilonnd{array}
\epsilonnd{equation}
Consider now $\Theta:=\Psi_2\circ\Psi\circ\Psi_1^{-1}$, defined in the open set $U_{c,\epsilonta,\varepsilon}$. $\Theta$ sends $\Psi_1(\widetilde{{\mathcal F}}_{{\Omegaega}_1})$ on $\Psi_2(\widetilde{{\mathcal F}}_{{\Omegaega}_2})$, and respects the first integral $$x^{m_{\nu}}s^{n_\nu+2(\varepsilonilon-1)}t^{n_\nu-2\varepsilonilon}.$$
It is defined on a set of the type
$${|x^{m_{\nu}}s^{n_\nu+2(\varepsilonilon-1)}t^{n_\nu-2\varepsilonilon}|<\varepsilon},$$ in the considered charts, and this set intersects the domain of definition of $\Psi$. So $\Psi=\Psi_2\circ\Theta\circ\Psi_1^{-1}$ can be extended to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}\cap D_{(\alpha-1)(\begin{enumerate}ta-1)}$.
Repeat the same argument to extend $\Psi$ to a neighbourhood of
$$\begin{itemize}g(D_{(\alpha-1)\begin{enumerate}ta}\cap D_{\alpha_2}\begin{itemize}g)\cup\Big(D_{(\alpha-1)\begin{enumerate}ta}\cap(\cup D_{(\alpha-2)j})\Big)\cup (D_{(\alpha-1)\begin{enumerate}ta}\cap \widetilde{S}),$$
and in consequence, $\Psi$ can be extended to a neighbourhood of $D_{(\alpha-1)\begin{enumerate}ta}$. This process can be repeated and extend $\Psi$ to a neighbourhood of the exceptional divisor $E$. So ${\mathcal F}_{\Omegaega_i}$ are analytically conjugated, outside of the singular locus of codimension two. Finally, using Hartogs theorem we can obtain the extension of the conjugation around of the origin.
In the case $d, q$ odd and $p$ even, we proceed analogously.
\epsilonnd{proof}
\section*{Acknowledgments}
The authors want to thank the Pontificia Universidad Cat\'{o}lica del Per\'{u} and the Universidad de Valladolid for their hospitality during the visits while preparing this paper.
The authors would like to thank the anonymous referee for many valuable and constructive suggestions, that have helped to improve the paper.
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\epsilonnd{thebibliography}
\epsilonnd{document} | math | 54,680 |
\begin{document}
\title[The quadratic covariation for a weighted-fBm]{The quadratic covariation for a weighted fractional Brownian motion${}^{*}$}
\footnote[0]{${}^{*}$The Project-sponsored by NSFC (No. 11571071, 11426036), Innovation Program of Shanghai Municipal Education
Commission (No. 12ZZ063) and Natural Science Foundation of Anhui
Province (No.1408085QA10)}
\author[X. Sun, L. Yan and Q. Zhang]{Xichao Sun${}^{\dag,\natural}$, Litan Yan${}^{\ddag,\S}$ and Qinghua Zhang${}^{\ddag}$}
\footnote[0]{${}^{\natural}[email protected], ${}^{\S}[email protected] (Corresponding Author)}
\date{}
\keywords{Weighted fractional Brownian motion, local time, Malliavin calculus, quadratic covariation, It\^{o} formula}
\subjclass[2000]{60G15, 60H05, 60G17}
\maketitle
\begin{center}
{\footnotesize {\it ${}^{\dag}$Department of Mathematics and Physics, Bengbu University\\
1866 Caoshan Rd., Bengbu 233030, P.R. China\\
${}^{\ddag}$Department of Mathematics, College of Science, Donghua University\\
2999 North Renmin Rd. Songjiang, Shanghai 201620, P.R. China}}
\end{center}
\maketitle
\begin{abstract}
Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property
$$
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0)
$$
for all $s>0$, we consider the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow
0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon}
\left\{f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds,
$$
provided the limit exists uniformly in probability. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\Omega)$ and the generalized Bouleau-Yor identity
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
holds for all $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)=\int_0^t\delta(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function.
\end{abstract}
\section{Introduction}
Long/short range dependence (or long/short memory) stochastic processes with self-similarity have been intensively used as models for different physical phenomena. These properties appeared in empirical studies in areas like hydrology and geophysics; and they appeared to play an important role in network traffic analysis, economics and telecommunications. As a consequence, some efficient mathematical models based on long/short range dependence processes with self-similarity have been proposed in these directions. We refer to the monographs of self-similar processes by Embrechts-Maejima~\cite{Embrechts-Maejima}, Sheluhin et al~\cite{Sheluhin et al.}, Samorodnitsky~\cite{Samorodnitsky}, Samorodnitsky-Taqqu~\cite{Samorodnitsky-Taqqu}, Taqqu~\cite{Taqqu3} and Tudor~\cite{Tudor}.
The fractional Brownian motion is a simple stochastic process with long/short range dependence and self-similarity which is a suitable generalization of standard Brownian motion. Some surveys and complete literatures on fractional Brownian motion could be found in Biagini {\it et al}~\cite{BHOZ}, Gradinaru et al.~\cite{Grad1}, Hu~\cite{Hu2}, Mishura~\cite{Mishura2}, Nualart~\cite{Nualart1}. On the other hand, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about self-similar Gaussian
processes and fields in general. Therefore, some generalizations of
the fBm has been introduced. However, contrast to the extensive
studies on fractional Brownian motion, there has been little
systematic investigation on other self-similar Gaussian processes.
The main reason for this is the complexity of dependence structures
for self-similar Gaussian processes which do not have stationary
increments. Thus, it seems interesting to study some extensions of fractional Brownian motion such as bi-fractional Brownian motion and the weighted fractional Brownian motion.
In this paper we consider the weighted fractional Brownian motion (weighted-fBm). Recall that the so-called weighted-fBm $B^{a,b}$ with parameters $a$ and $b$ is a zero mean Gaussian process with long/short-range dependence and self-similarity. It admits the relatively simple covariance as follows
$$
E\left[B^{a,b}_tB^{a,b}_s\right]=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du
$$
where ${\mathbb B}(\cdot,\cdot)$ is the beta function and
$a>-1,|b|<1,|b|<a+1$. Clearly, if $a=0$, the process coincides with
the standard fractional Brownian motion with Hurst parameter
$H=\frac{b+1}2$, and it admits the explicit significance. We have
(see, Lemma~\ref{lem3.0} in Section~\ref{sec3}, see also Bojdecki
{\em et al}~\cite{Bojdecki1})
\begin{equation}\label{sec1-eq1-1}
c_{a,b}(t\vee s)^a|t-s|^{b+1}\leq
E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\leq C_{a,b} (t\vee
s)^a|t-s|^{b+1}
\end{equation}
for $s,t\geq 0$. Thus, Kolmogorov's continuity criterion implies
that weighted-fractional Brownian motion is $\gamma$-H\"{o}lder
continuous for any $\gamma<\frac{1+b}2$, where $\frac{1+b}2$ is
called the H\"older continuous index. The process $B^{a,b}$ is $\frac12(a+b+1)$-self similar and its increments are not stationary. It is important to note that the following fact:
\begin{itemize}
\item The H\"older continuous index $\frac12(1+b)$ is not equal either to the its self-similar index nor the order of the infinitesimal $\sqrt{E[(X_t)^2]}\to 0$ as $t\downarrow 0$, provided $a\neq 0$.
\end{itemize}
However, the three indexes are coincident for many famous self-similar Gaussian processes such as fractional Brownian motion,
sub-fractional Brownian motion and bi-fractional Brownian motion.
That is causing trouble for the research, and it is also our a motivation to study the weighted-fBm. Before making the decision to study the weighted-fBm we first try to investigate in Yan et al.~\cite{Yan3} some path properties including strong local nondeterminism, Chung's law of the iterated logarithm and the smoothness of the collision local time. In particular, we showed that it is strongly locally $\phi$-nondeterministic with $\phi(r)=r^{1+b}$. In general, the function $\phi$ depends on the self-similar index of the process, but the fact is, for the weighted-fBm, $\phi(r)=r^{1+b}$ is independent of parameter $a$, which enhances further our interesting to study the weighted-fBm.
The weighted-fBm appeared in Bojdecki {\em et al}~\cite{Bojdecki1} in a limit of occupation time fluctuations of a system of independent particles moving in ${\mathbb R}^d$ according a symmetric $\alpha$-stable L\'evy process, $0<\alpha\leq 2$, started from an inhomogeneous
Poisson configuration with intensity measure
$$
\frac{dx}{1+|x|^\gamma}
$$
and $0<\gamma\leq d=1<\alpha$, $a=-\gamma/\alpha$, $b=1-1/\alpha$,
the ranges of values of $a$ and $b$ being $-1< a < 0$ and $0 < b\leq
1+a$. The process also appears in Bojdecki {\em et
al}~\cite{Bojdecki2} in a high-density limit of occupation time
fluctuations of the above mentioned particle system, where the
initial Poisson configuration has finite intensity measure, with
$d=1<\alpha$, $a=-1/\alpha$, $b=1-1/\alpha$. Moreover, the
definition of the weighted-fBm $B^{a,b}$ was first introduced by
Bojdecki {\em et al}~\cite{Bojdecki0}, and it is neither a
semimartingale nor a Markov process if $b\neq 0$, so many of the
powerful techniques from stochastic analysis are not available when
dealing with $B^{a,b}$. There has been little systematic investigation on weighted-fBm since it it has been introduced by Bojdecki {\em et al}~\cite{Bojdecki0}.
In this paper, we consider the the {\it generalized quadratic covariation} when $b<0$, and it is important to note that a large class of Gaussian processes with similar characteristics as weighted-fBm could be handled in uniform approach used here. Clearly, by the estimates~\eqref{sec1-eq1-1}, we have
$$
E\bigl[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2\bigr]=O( s^a\varepsilon^{1+b})\qquad (\varepsilon\to 0)
$$
for all $s>0$, which implies that
$$
\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon} \int_{t_0}^{t}(B_{s+\varepsilon}-B_s)^2ds=
\begin{cases}
0, & \text{ {if $0<b<1$}}\\
+\infty,& \text{ {if $-1<b<0$}}
\end{cases}
$$
for all $t\geq t_0>0$, where the limit is uniformly in probability. Additional results on the quadratic variation can be found in Russo-Vallois~\cite{Russo2}. Thus, we need a substitute tool of the quadratic variation for $b\neq 0$. Inspired by~\eqref{sec1-eq1-1}, the fact
$$
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0)
$$
for all $s>0$ and Cauchy's principal value, one can naturally give the following definition.
\begin{definition}
Let $a>-1,|b|<1,|b|<a+1$ and let the integral
$$
J_\varepsilon(f,t):=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
exists for all $\varepsilon>0$ and all Borel functions $f$. The limit
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t:=\lim_{\varepsilon\downarrow 0}J_\varepsilon(f,t)
$$
is called the {\it generalized quadratic covariation} of $f(B^{a,b})$ and $B^{a,b}$, provided the limit exists uniformly in probability.
\end{definition}
\begin{remark}\label{rem}
{\rm
It is important to note that the above definition is available for a large class of Gaussian processes with similar characteristics as weighted-fBm. Let now $X$ be a self-similar Gaussian process with H\"older continuous paths of order $\alpha\in (0,1)$. We then can define the generalized quadratic covariation $[f(X),X]^{(a,b)}$ as follows
$$
[f(X),X]^{(a,b)}_t=\lim_{\varepsilon\downarrow
0}\frac{2\alpha}{\varepsilon^{2\alpha}}\int_\varepsilon^{t+\varepsilon}
\left\{f(X_{s+\varepsilon})
-f(X_s)\right\}(X_{s+\varepsilon}-X_s)s^{2\alpha-1}ds
$$
for any Borel functions $f$, provided the limit exists uniformly in probability. When $0<\alpha<\frac12$, we can get some similar results for the process $X$ to weighted-fBm with $-1<b<0$.
}
\end{remark}
We shall see in Section~\ref{sec6} that
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t =\kappa_{a,b}\int_0^tf'(B^{a,b}_s)ds^{1+a+b},\qquad t\geq 0
$$
for all $f\in C^1({\mathbb R})$ and all $b\in (-1,1)$, where
$$
\kappa_{a,b}=\frac1{(1+b){\mathbb B}(a+1,b+1)}.
$$
In the present paper we prove the existence of the generalized quadratic covariation for $-1<b<0$, our start point is to consider the decomposition
\begin{equation}\label{sec1-eq1-2}
\begin{split}
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} &\left\{f(B^{a,b}_{
s+\varepsilon})-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)s^{b}ds\\
&=\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_{
s+\varepsilon})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\ &\qquad\qquad-\frac{1}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds
\end{split}
\end{equation}
for all $-1<b<0$. It is important to note that the above decomposition is unavailable for $0<b<1$. For example, we have
$$
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} E\left(B^{a,b}_{s} (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right)s^bds\longrightarrow \infty
$$
for all $b>0$ and $t>0$, as $\varepsilon$ tends to zero, because
\begin{equation}\label{sec1-eq1-3}
E\left(B^{a,b}_{s} (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right) \sim s^{a+b}\varepsilon\qquad (\varepsilon\to 0).
\end{equation}
The above asymptotic property follows from
\begin{align*}
E[B^{a,b}_s(B^{a,b}_{s+\varepsilon}&-B^{a,b}_s)]=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a[({s+\varepsilon}-u)^b+(s-u)^b]du-s^{1+a+b}\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a\left[({s+\varepsilon}-u)^b-(s-u)^b\right]du\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a({s+\varepsilon}-u)^b\left[1- (\frac{s-u}{{s+\varepsilon}-u})^b\right]du\\
&\sim s^{a+b}\varepsilon
\end{align*}
for all $b>0$ by the fact $1-x^b\sim 1-x$ as $x\to 1$. Thus, the method used here is different from that need to handle the case $0<b<1$.
This paper is organized as follows. In Section~\ref{sec2} we present some preliminaries for weighted-fBm and Malliavin calculus. In Section~\ref{sec3}, we establish some technical estimates associated with weighted-fBm with $-1<b<0$. In Section~\ref{sec6}, as an example, when $f\in C^1({\mathbb R})$ we show that the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ exists in $L^2$ for all $a,b$ and
$$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\kappa_{a,b}\int_0^t f'(B^{a,b}_s)ds^{1+a+b}.
$$
In particular, we have
$$
\bigl[B^{a,b},B^{a,b}\bigr]^{(a,b)}_t=\kappa_{a,b}t^{1+a+b}.
$$
In Section~\ref{sec5}, in more general cases we consider the existence of generalized quadratic covariation for $-1<b<0$. By estimating the two terms of the right hand side in the decomposition~\eqref{sec1-eq1-2}, respectively, we construct a Banach space ${\mathscr H}$ of measurable functions $f$ on
${\mathbb R}$ such that $\|f\|_{{\mathscr H}}<\infty$, where
\begin{align*}
(\|f\|_{{\mathscr H}})^2:=\int_0^{T+1}\int_{\mathbb
R}|f(x)|^2e^{-\frac{x^2}{2s^{1+a+b}
}}\frac{dxds}{\sqrt{2\pi}s^{(1-a-b)/2}}.
\end{align*}
We show that the {\it generalized quadratic covariation} $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ exists for all $f\in {\mathscr H}$ and
$$
E\left|\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\right|^2\leq C_{a,b,T}\|f\|_{\mathscr H}^2,\qquad 0\leq t\leq T.
$$
In Section~\ref{sec7}, for $-1<b<0$ we consider the integral
\begin{equation}\label{sec1-eq1-4}
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{equation}
for $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)$ denotes the weighted local time defined by
$$
{\mathscr L}^{a,b}(x,t)=(1+a+b)\int_0^t\delta(B^{a,b}_s-x)s^{a+b}ds.
$$
In order to study the integral we obtain the following It\^o formula :
$$
F(B^H)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t,
$$
where $F$ is an absolutely continuous function with $F'=f\in {\mathscr H}$, and show that the {\em generalized Bouleau-Yor identity}
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\kappa_{a,b}\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
holds for all $f\in {\mathscr H}$. As a corollary we get the Tanaka formula
$$
|B^{a,b}_t-x|=|-x|+\int_0^t{\rm sign}(B^{a,b}_s-x)dB^{a,b}_s+{\mathscr L}^{a,b}(x,t)
$$
for $-1<b<0$.
\section{The weighted fractional Brownian motion}\label{sec2}
Let $B^{a,b}$ be a weighted-fBm with parameters $a,b$
($a>-1,|b|<1,|b|<a+1$), defined on the complete probability space
$(\Omega,\mathcal{F},P)$. As we pointed out before, the weighted-fBm
$B^{a,b}=\left\{B^{a,b}_t,0\leq t\leq T \right\}$ with indices $a$
and $b$ is a mean zero Gaussian processes such that $B_0^{a,b}=0$ and
\begin{equation}\label{sec2-eq2.1}
E\left[B^{a,b}_tB^{a,b}_s\right]=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du
\end{equation}
for $s,t\geq 0$. It is known that the function $(t,s)\mapsto
R^{a,b}(t,s)$ is positive-definite if and only if $a$ and $b$
satisfy the conditions
\begin{equation}\label{sec2-eq2.2}
a>-1,\;|b|<1,\;|b|<a+1,
\end{equation}
and the following statements hold (see Bojdecki~\cite{Bojdecki0}):
\begin{itemize}
\item $B^{a,b}$ is $\frac12(a+b+1)$-self similar;
\item $B^{a,b}$ has independent increments for $b=0$;
\item $B^{a,b}$ is neither a semimartingale nor a Markov process if $b\neq 0$;
\item If $b>0$, then $B^{a,b}$ is long-range dependence;
\item If $b<0$, then $B^{a,b}$ is short-range dependence.
\end{itemize}
Thus, throughout this paper we let $b\neq 0$ for simplicity.
As a Gaussian process, it is possible to construct a stochastic calculus of variations with respect to $B^{a,b}$. We refer to Al\'os {\it et al}~\cite{Nua1} and Nualart~\cite{Nualart1} for the complete descriptions of stochastic calculus with respect to Gaussian processes. Here we recall only the basic elements of this theory. Throughout this paper we assume that~\eqref{sec2-eq2.2} holds.
Let ${\mathcal H}_{a,b}$ be the completion of the linear space ${\mathcal
E}$ generated by the indicator functions $1_{[0,t]}, t\in [0,T]$
with respect to the inner product
$$
\langle 1_{[0,s]},1_{[0,t]} \rangle_{{\mathcal H}_{a,b}}=R^{a,b}(s,t)=\frac{1}{2{\mathbb
B}(a+1,b+1)}\int_0^{s\wedge t}u^a((t-u)^b+(s-u)^b)du.
$$
The application $\varphi\in {\mathcal E}\to B(\varphi)$ is an
isometry from ${\mathcal E}$ to the Gaussian space generated by $B^{a,b}$ and it can be extended to ${\mathcal H}_{a,b}$.
\begin{remark}
{\rm
For $b>0$ we can characterize ${\mathcal H}_{a,b}$ as
$$
{\mathcal H}_{a,b}=\{f:[0,T]\to {\mathbb R}\;|\;\|f\|_{a,b}<\infty\},
$$
where
$$
\|f\|^2_{a,b}=\int_0^T\int_0^Tf(t)f(s)\frac{\partial^2}{\partial s\partial t}R^{a,b}(s,t)dsdt.
$$
Clearly, we can write its covariance as
\begin{align*}
\phi_{a,b}(s,t):=\frac{\partial^2}{\partial t\partial s}R^{a,b}(t,s) &=\frac{b}{2{\mathbb
B}(a+1,b+1)}(t\wedge s)^a|t-s|^{b-1}
\end{align*}
for $b>0$. Thus, $R^{a,b}$ is the distribution function of an absolutely continuous positive measure with density $\frac{b}{2{\mathbb
B}(a+1,b+1)}(t\wedge s)^a|t-s|^{b-1}$ which belongs of course to
$L^1([0,T ]^2)$.
}
\end{remark}
Let us denote by ${\mathcal S}^{a,b}$ the set of smooth functionals of the form
$$
F=f(B(\varphi_1),B(\varphi_2),\ldots,B(\varphi_n)),
$$
where $f\in C^{\infty}_b({\mathbb R}^n)$ ($f$ and all its derivatives are bounded) and $\varphi_i\in {\mathcal H}_{a,b}$. The {\em Malliavin derivative} $D^{a,b}$ of a functional $F$ as above is given by
$$
D^{a,b}F=\sum_{j=1}^n\frac{\partial f}{\partial
x_j}(B(\varphi_1),B(\varphi_2), \ldots,B(\varphi_n))\varphi_j.
$$
The derivative operator $D^{a,b}$ is then a closable operator from $L^2(\Omega)$ into $L^2(\Omega;{\mathcal H}_{a,b})$. We denote by ${\mathbb D}^{1,2}$ the closure of ${\mathcal S}_{a,b}$ with respect to the norm
$$
\|F\|_{1,2}:=\sqrt{E|F|^2+E\|D^{a,b}F\|^2_{a,b}}.
$$
The {\it divergence integral} $\delta^{a,b}$ is the adjoint of derivative operator $D^{a,b}$. That is, we say that a random variable $u$ in $L^2(\Omega;{\mathcal H}_{a,b})$ belongs to the domain of the divergence operator $\delta^{a,b}$, denoted by ${\rm {Dom}}(\delta^{a,b})$, if
$$
E\left|\langle D^{a,b}F,u\rangle_{{\mathcal H}_{a,b}}\right|\leq
c\|F\|_{L^2(\Omega)}
$$
for every $F\in {\mathbb D}^{1,2}$, where $c$ is a constant depending only on $u$. In this case $\delta^{a,b}(u)$ is defined by the duality relationship
\begin{equation}
E\left[F\delta^{a,b}(u)\right]=E\langle D^{a,b}F,u\rangle_{{\mathcal H}_{a,b}}
\end{equation}
for any $F\in {\mathbb D}^{1,2}$, we have ${\mathbb D}^{1,2}\subset
{\rm {Dom}}(\delta^{a,b})$. We will denote
$$
\delta^{a,b}(u)=\int_0^Tu_sdB^{a,b}_s
$$
for an adapted process $u$, and it is called Skorohod integral. We have the following It\^o formula.
\begin{theorem}[Al\'os {\it et al}~\cite{Nua1}]\label{theorem-Ito}
Let $f\in C^{2}({\mathbb R})$ such that
\begin{equation}\label{sec2-Ito-con1}
\max\left\{|f(x)|,|f'(x)|,|f''(x)|\right\}\leq \kappa e^{\beta x^2},
\end{equation}
where $\kappa$ and $\beta$ are positive constants with
$\beta<\frac14T^{-(1+a+b)}$. Then we have
\begin{align*}
f(B^{a,b}_t)=f(0)&+\int_0^t\frac{d}{dx}f(B^{a,b}_s)dB^{a,b}_s
+\frac12(1+a+b)\int_0^t\frac{d^2}{dx^2}f(B^{a,b}_s)s^{a+b}ds
\end{align*}
for all $t\in [0,T]$.
\end{theorem}
\section{Some basic estimates}\label{sec3}
For simplicity throughout this paper we let $C$ stand for a positive constant depending only on the subscripts and its value may be different in different appearance, and this assumption is also adaptable to $c$. Moreover, the notation $F\asymp G$ means that there are positive constants $c_1$ and $c_2$
so that
$$
c_1G(x)\leq F(x)\leq c_2G(x)
$$
in the common domain of definition for $F$ and $G$. For $x,y\in
\mathbb{R}$, $x\wedge y:=\min\{x,y\}$ and $x\vee y:=\max\{x,y\}$.
\begin{lemma}\label{lem3.0}
Let $a>-1,\;|b|<1,\;|b|<a+1$. We then have
\begin{equation}\label{sec3-eq3.1}
Q(t,s):=E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\asymp (t\vee
s)^a|t-s|^{1+b}
\end{equation}
for $s,t\geq 0$. In particular, we have
\begin{equation}\label{sec3-eq3.2}
E\left[\left(B^{a,b}_t-B^{a,b}_s\right)^2\right]\leq
C_{a,b}|t-s|^{1+a+b}
\end{equation}
for $a\leq 0$.
\end{lemma}
The estimates~\eqref{sec3-eq3.1} are first considered by
Bojdecki~\cite{Bojdecki0}. The present form is a slight
modification given by Yan et al.~\cite{Yan3}. Thus, Kolmogorov's continuity criterion and the Gaussian property of the process imply that
weighted-fractional Brownian motion is $\gamma$-H\"{o}lder
continuous for any $\gamma<\frac{1+b}2$, where $\frac{1+b}2$ is
called the {\em H\"older continuity index}. It is important to note
that the H\"older continuity index $\frac{1+b}2$ is not equal to the
order $\frac{1+a+b}2$ of the infinitesimal $\sqrt{E[(X_t)^2]}\to 0$
as $t\downarrow 0$ unless $a=0$. However, the H\"{o}lder continuity
index of many popular self-similar Gaussian processes equals to the
order of the infinitesimal such as fractional Brownian motion,
sub-fractional Brownian motion and bi-fractional Brownian motion.
\begin{lemma}\label{lem3.1}
Let $a>-1,\;|b|<1,\;|b|<a+1$. We then have
\begin{equation}\label{sec3-eq3.3}
t^{1+a+b}s^{1+a+b}-\mu^2\asymp (ts)^{a}(t\wedge s)^{1+b}|t-s|^{1+b}
\end{equation}
for all $s,t>0$, where $\mu=E(B^{a,b}_tB^{a,b}_s)$.
\end{lemma}
\begin{proof}
Without loss of generality we may assume that $t>s>0$. Then
\begin{align*}
t^{1+a+b}&s^{1+a+b}-\mu^2\\
&=t^{1+a+b}s^{1+a+b}-\frac1{4{\mathbb B}^2(1+a,1+b)}\left(\int_0^su^a\left((t-u)^b+(s-u)^b\right)du\right)^2\\
&\equiv t^{2(1+a+b)}G(x)
\end{align*}
with $x=\frac{s}{t}$, where
$$
G(x)=x^{1+a+b}-\frac1{4{\mathbb B}^2(1+a,1+b)}\left(\int_0^xu^a(1-u)^bdu+{\mathbb B}(1+a,1+b)x^{1+a+b}\right)^2
$$
with $x\in [0,1]$. Noting that
$$
t^{1+a+b}s^{1+a+b}-\mu^2\geq 0
$$
and
$$
t^{1+a+b}s^{1+a+b}-\mu^2=0\qquad\Longleftrightarrow\qquad s=t\quad {\text {or }}\quad s=0
$$
for all $t\geq s>0$, we see that
$G(x)\geq 0$ and
$$
G(x)=0\qquad \Longleftrightarrow \qquad x=0\quad{\text {or }}\quad x=1.
$$
Decompose $G(x)$ as follows
\begin{align*}
G(x)&=\frac1{4(K_{a,b})^2}\left(2K_{a,b} x^{\frac{1+a+b}2}-\int_0^xu^a(1-u)^bdu-K_{a,b}x^{1+a+b}\right)\\
&\qquad\quad\cdot\left(2K_{a,b} x^{\frac{1+a+b}2}+\int_0^xu^a(1-u)^bdu+K_{a,b}x^{1+a+b}\right)\\
&\equiv \frac1{4(K_{a,b})^2}G_1(x)G_2(x)
\end{align*}
for all $x\in [0,1]$, where $K_{a,b}={\mathbb B}(1+a,1+b)$.
Obviously, we have
$$
K_{a,b} x^{\frac{1+a+b}2}\leq G_2(x)\leq C_{a,b}x^{\frac{1+a+b}2}
$$
for all $x\in [0,1]$. In fact, the left inequality is clear, and the right inequality follows from
the fact
$$
\int_0^xv^a(1-v)^bdv\leq \int_0^xv^a\left((x-v)^b+1\right)dv\leq K_{a,b}x^{1+a+b}+\frac1{1+a}x^{1+a}
$$
for all $x\in [0,1]$. On the other hand, we also have
\begin{align*}
\lim_{x\uparrow 1}\frac{G_1(x)}{x^{\frac{1+a+b}2}(1-x)^{1+b}}=\frac1{1+b},\qquad \lim_{x\downarrow 0}\frac{G_1(x)}{x^{\frac{1+a+b}2}(1-x)^{1+b}}=2{\mathbb B}(1+a,1+b),
\end{align*}
which deduces
$$
G_1(x)\asymp x^{\frac{1+a+b}2}(1-x)^{1+b}
$$
by the continuity. Thus, we have showed that
$$
G(x)=G_1(x)G_2(x)\asymp x^{1+a+b}(1-x)^{1+b}
$$
and the lemma follows.
\end{proof}
\begin{lemma}\label{lem3.3}
Let $t>s>t'>s'>0$ and let $-1<b<1$, $a>-1$, $|b|<1+a$. We then have
\begin{equation}\label{sec3-eq3.6}
\begin{split}
|E(B^{a,b}_{t}-&B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})|
\\
&\leq C_{a,b,\alpha}\left((s')^a\vee s^a\right)^\alpha(tt')^{\frac12a(1-\alpha)} \frac{[(t-s)(t'-s')]^{\alpha+\frac12(1-\alpha)(1+b)} }{(t-t')^{(1-b)\alpha}}
\end{split}
\end{equation}
for all $\alpha\in [0,1]$.
\end{lemma}
\begin{proof}
Let $b<0$. Denote
\begin{align*}
\mu(t,s,t',s'):&=E(B^{a,b}_{t}-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_{s'}^{t'}u^a\left[(s-u)^b-(t-u)^b\right]du.
\end{align*}
It follows from the fact
\begin{equation}\label{sec3-eq3.7=0}
\begin{split}
y^\gamma-x^\gamma&=y^\gamma\left(1-(\frac{x}{y})^\gamma\right)\leq C_{\gamma}y^\gamma\left(1-\frac{x}{y}\right)\\
&\leq C_{\gamma,\beta}y^\gamma\left(1-\frac{x}{y}\right)^\beta
\leq C_{\gamma,\beta}y^{\gamma-\beta}(y-x)^\beta
\end{split}
\end{equation}
for $y>x>0$, $\gamma\geq 0$, $0\leq \beta\leq 1$ and the inequality
\begin{align*}
t-u=(t-t')+(t'-u)\geq (t-t')^{1-\nu}(t'-u)^\nu\qquad (0<u<t'<t)
\end{align*}
for all $0\leq \nu\leq 1$ that
\begin{equation}\label{sec3-eq3.8==}
\begin{split}
|\mu(t,s,&t',s')|=\frac1{2{\mathbb B}(1+a,1+b)}\int_{s'}^{t'}u^a\left[(s-u)^b -(t-u)^b\right]du\\
&\leq C_{a,b}\int_{s'}^{t'}u^a\frac{(t-u)^{-b} -(s-u)^{-b}}{
(s-u)^{-b}(t-u)^{-b}}du\leq C_{a,b,\beta}\int_{s'}^{t'}u^a\frac{(t-s)^\beta}{
(s-u)^{-b}(t-u)^\beta}du\\
&\leq C_{a,b}\frac{(t-s)^\beta}{(t-t')^{\beta(1-\nu)}} \int_{s'}^{t'}\frac{u^a}{(t'-u)^{-b+\beta\nu}}du\\
&\leq C_{a,b}\left[(s')^a\vee s^a\right]\frac{(t-s)^\beta(t'-s')^{1+b-\beta\nu}}{(t-t')^{\beta(1-\nu)}}
\end{split}
\end{equation}
for all $0\leq \beta\leq 1$ and $0\leq \nu\beta\leq 1+b$.
On the other hand, noting that
\begin{align*}
|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|^2&\leq E\left[(B^{a,b}_t-B^{a,b}_s)^2\right]E\left[(B^{a,b}_{t'} -B^{a,b}_{s'})^2\right]\\
&\leq C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b},
\end{align*}
we see that
\begin{align*}
&\frac{|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|}{ {\sqrt{C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b}}}}\leq
\left(\frac{|E[(B^{a,b}_t-B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|}{ {\sqrt{C_{a,b}(tt')^a(t-s)^{1+b}(t'-s')^{1+b}}}}\right)^\alpha
\end{align*}
for all $\alpha\in [0,1]$. Combining this with~\eqref{sec3-eq3.8==} (taking $\beta=1+b$ and $\nu=0$), we get
\begin{align*}
|E[(B^{a,b}_t-&B^{a,b}_s)(B^{a,b}_{t'}-B^{a,b}_{s'})]|\\
&\leq
C_{a,b,\alpha}\left((s')^a\vee s^a\right)^\alpha(tt')^{\frac12a(1-\alpha)} \frac{[(t-s)(t'-s')]^{\alpha+\frac12(1-\alpha)(1+b)} }{(t-t')^{(1+b)\alpha}}
\end{align*}
and the lemma follows for all $\alpha\in [0,1]$. Similarly, we can show that the lemma holds for $b>0$.
\end{proof}
\begin{lemma}\label{lem3.4}
For $a>-1$, $-1<b<0$ and $|b|<1+a$ we have
\begin{align}\label{lem3.4-eq1}
&E[B^{a,b}_t(B^{a,b}_s-B^{a,b}_r)]\leq C_{a,b}(s-r)^{1+b}s^a\\ \label{lem3.4-eq2}
&E[B^{a,b}_s(B^{a,b}_t-B^{a,b}_s)]\leq C_{a,b}(t-s)^{1+b}s^a\\ \label{lem3.4-eq3}
&E[B^{a,b}_s(B^{a,b}_s-B^{a,b}_r)]\leq C_{a,b}(s-r)^{1+b}s^a\\ \label{lem3.4-eq4}
&E[B^{a,b}_{s}(B^{a,b}_t-B^{a,b}_r)]\leq C_{a,b}(t-r)^{1+b}s^a\\ \label{lem3.4-eq5}
&E[B^{a,b}_{r}(B^{a,b}_t-B^{a,b}_s)]\leq C_{a,b}(t-s)^{1+b}r^a
\end{align}
for all $t>s>r>0$.
\end{lemma}
\begin{proof}
Let $t>s>r>0$. By~\eqref{sec3-eq3.7=0} we have
\begin{equation}\label{lem3.4-eq6}
s^{1+a+b}-r^{1+a+b}\leq C_{a,b}s^{a+b}(s-r)\leq C_{a,b}(s-r)^{1+b}s^a
\end{equation}
for all $a>-1$, $-1<b<0$ and $0\leq \beta\leq 1$. Notice that
\begin{equation}\label{lem3.1-eq3}
\int_{x}^1r^a(1-r)^bdr \asymp (1-x)^{1+b}
\end{equation}
for all $x\in [0,1]$ by the continuity and the convergence
\begin{align*}
\lim_{x\to 1}\frac{f(x)}{(1-x)^{1+b}}=\frac1{1+b}
\end{align*}
for all $a,b>-1$. We get
\begin{align*}
\int_r^su^a(t-u)^bdu&\leq \int_r^su^a(s-u)^bdu=s^{1+a+b}\int_{r/s}^1v^a(1-v)^bdv\\
&\leq C_{a,b}s^{1+a+b}(1-\frac{r}s)^{1+b} =C_{a,b}(s-r)^{1+b}s^a
\end{align*}
for $-1<b<0,a>-1$. It follows that
\begin{align*}
E[B^{a,b}_t&(B^{a,b}_s-B^{a,b}_r)]=\frac1{2{\mathbb B}(1+a,1+b)}\left(\int_0^su^a[(t-u)^b+(s-u)^b]du\right.\\
&\hspace{3cm}\left.-
\int_0^ru^a[(t-u)^b+(r-u)^b]du\right)\\
&=\frac1{2{\mathbb B}(1+a,1+b)}\int_r^su^a(t-u)^bdu +\frac12\left(s^{1+a+b}-r^{1+a+b}\right)\\
&\leq C_{a,b}(s-r)^{1+b}s^a.
\end{align*}
This establishes the estimate~\eqref{lem3.4-eq1}. In order to prove the estimate~\eqref{lem3.4-eq2} we have
\begin{equation}\label{lem3.4-eq7}
\int_0^{x}v^a(1-v)^bdv\asymp x^{1+a}
\end{equation}
for all $x\in [0,1]$ by the continuity and the convergence
$$
\lim_{x\downarrow 0}\frac{1}{x^{1+a}}\int_0^xv^a(1-v)^bdv=\frac1{1+a}
$$
for all $a,b>-1$. It follows that
\begin{align*}
|E[B^{a,b}_s(B^{a,b}_t&-B^{a,b}_s)]|=\left|\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a[(t-u)^b+(s-u)^b]du-s^{1+a+b}\right|\\
&=\left|\frac1{2{\mathbb B}(1+a,1+b)}\int_0^su^a(t-u)^bdu-\frac12s^{1+a+b}\right|\\
&=\left|\frac{t^{1+a+b}}{2{\mathbb B}(1+a,1+b)}\int_0^{s/t}v^a(1-v)^bdv-\frac12s^{1+a+b}\right|\\
&\leq \frac{1}{2{\mathbb B}(1+a,1+b)}\int_0^{s/t}v^a(1-v)^bdv\left(t^{1+a+b}-s^{1+a+b}\right)\\
&\qquad +s^{1+a+b}\frac{1}{2{\mathbb B}(1+a,1+b)}\left({\mathbb B}(1+a,1+b)-\int_0^{s/t}v^a(1-v)^bdv\right)\\
&\leq C_{a,b} (t-s)^{1+b}s^a,
\end{align*}
which gives~\eqref{lem3.4-eq2}. Similarly, we can prove the estimate~\eqref{lem3.4-eq3}. The estimate~\eqref{lem3.4-eq4} follows from~\eqref{lem3.4-eq2} and~\eqref{lem3.4-eq3}.
Finally, in order to prove~\eqref{lem3.4-eq5} we have
\begin{align*}
|E[B^{a,b}_{r}&(B^{a,b}_t-B^{a,b}_s)]|=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^ru^a[(s-u)^b-(t-u)^b]du\\
&\leq \frac{r^a}{2{\mathbb B}(1+a,1+b)}\int_0^r[(s-u)^b-(t-u)^b]du\\
&\leq \frac{r^a}{2{\mathbb B}(1+a,1+b)}\int_0^s[(s-u)^b-(t-u)^b]du\\
&=C_{a,b}r^a\left(s^{1+b}-t^{1+b}+(t-s)^{1+b}\right)\leq C_{a,b}(t-s)^{1+b}r^a
\end{align*}
for all $a\geq 0$ and $-1<b<0$. Moreover, we also have
\begin{align*}
s^{1+a+b}\Bigl(\int_0^{r/s}& v^a(1-v)^bdv -\int_0^{r/t}v^a(1-v)^bdv\Bigr) =s^{1+a+b}\int_{r/t}^{r/s}v^a(1-v)^bdv\\
&\leq \frac{s^{1+a+b}}{1+b}(\frac{r}t)^a \left(\frac{r}s-\frac{r}{t}\right)^{1+b}\\
&=\frac{s^{1+a+b}}{1+b}(\frac{r}t)^a \frac{(t-s)^{1+b}r^{1+b}}{(ts)^{1+b}}\leq \frac{1}{1+b}r^a (t-s)^{1+b}
\end{align*}
for $-1<a<0$. It follows from~\eqref{lem3.4-eq7} that
\begin{align*}
|E[B^{a,b}_{r}&(B^{a,b}_t-B^{a,b}_s)]|=\frac1{2{\mathbb B}(1+a,1+b)}\int_0^ru^a[(s-u)^b-(t-u)^b]du\\
&=C_{a,b}\left(s^{1+a+b}
\int_0^{r/s}v^a(1-v)^bdv-t^{1+a+b}\int_0^{r/t}v^a(1-v)^bdv\right)\\
&\leq C_{a,b}
s^{1+a+b}
\left(\int_0^{r/s}v^a(1-v)^bdv-\int_0^{r/t}v^a(1-v)^bdv\right)\\
&\qquad +C_{a,b}\int_0^{r/t}v^a(1-v)^bdv
\left(t^{1+a+b}-s^{1+a+b}\right)\\
&\leq C_{a,b} (t-s)^{1+b}r^a+C_{a,b}(\frac{r}{t})^{1+a} \left(t^{1+a+b}-s^{1+a+b}\right)\\
&\leq C_{a,b}(t-s)^{1+b}r^a,
\end{align*}
and the lemma follows.
\end{proof}
Let $\varphi_{t,s}(x,y)$ denote the density function of $(B^{a,b}_t,B^{a,b}_s)$ ($t>s>0$). That is
\begin{equation}\label{sec4-eq4.2}
\varphi_{t,s}(x,y)=\frac1{2\pi\rho_{t,s}}\exp\left\{
-\frac{1}{2\rho^2_{t,s}}\left(s^{1+a+b}x^2-2\mu_{t,s}
xy+t^{1+a+b}y^2\right)\right\},
\end{equation}
where $\mu_{t,s}=E(B^{a,b}_tB^{a,b}_s)$, $E\left[(B^{a,b}_t)^2\right]=t^{1+a+b}$ and
$\rho^2_{t,s}=(ts)^{1+a+b}-\mu_{t,s}^2$.
\begin{lemma}\label{lem3.5}
Let $a>-1,\;|b|<1,\;|b|<a+1$. If $f\in C^1({\mathbb R})$ admits compact support, we then have
\begin{align}\label{lemma3.5-1}
|Ef'(B^{a,b}_{s})f'(B^{a,b}_{r})|&\leq
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(E[f^2(B^{a,b}_s)]
+E[f^2(B^{a,b}_r)]\right)\\ \label{lemma3.5-2}
|Ef''(B^{a,b}_{s})f(B^{a,b}_{r})|&\leq
\frac{r^{1+a+b}}{\rho^2_{s,r}}\left(E[f^2(B^{a,b}_s)]
+E[f^2(B^{a,b}_r)]\right)
\end{align}
for all $s,r>0$.
\end{lemma}
\begin{proof}
Elementary computation shows that
\begin{align*}
\int_{\mathbb{R}^2}f^2(y)&(x-\frac{\mu_{t,s}}{r^{1+a+b}}
y)^2\varphi_{s,r}(x,y)dxdy\\
&=\frac{\rho^2_{s,r}}{r^{1+a+b}}\int_{\mathbb{R}}f^2(y)
\frac{1}{\sqrt{2\pi}r^{(1+a+b)/2}}
e^{-\frac{y^2}{2r^{1+a+b}}}dy=\frac{\rho^2_{s,r}}{r^{1+a+b}}
E|f(B^{a,b}_r)|^2,
\end{align*}
which implies that
\begin{align*}
\frac1{\rho^4_{s,r}}\int_{\mathbb{R}^2}|f(x)f(y)
(&s^{1+a+b}y-\mu_{s,r}x)(r^{1+a+b}x-\mu_{s,r}y)|\varphi_{s,r}(x,y)dxdy\\
&\leq \frac{r^{(1+a+b)/2}s^{(1+a+b)/2}}{\rho^2_{s,r}}\left(
E|f(B^{a,b}_s)|^2E|f(B^{a,b}_r)|^2\right)^{1/2}
\end{align*}
for all $s,r>0$. It follows that
\begin{align*}
|E&[f'(B^{a,b}_{s})f'(B^{a,b}_{r})]|=|\int_{\mathbb{R}^2}
f(x)f(y)\frac{\partial^{2}}{\partial x\partial
y}\varphi_{s,r}(x,y)dxdy|\\
&=|\int_{\mathbb{R}^2} f(x)f(y)\left\{\frac1{\rho^4_{s,r}}(s^{1+a+b}y-\mu_{s,r}x)(r^{1+a+b}x -\mu_{s,r}y)+\frac{\mu_{s,r}}{\rho^2_{s,r}}\right\}\varphi_{s,r}(x,y)dxdy|\\
&\leq \left(\frac{(rs)^{(1+a+b)/2}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(
E|f(B^{a,b}_s)|^2E|f(B^{a,b}_r)|^2\right)^{1/2}.
\end{align*}
Similarly, one can show that the estimate~\eqref{lemma3.5-2} holds.
\end{proof}
\section{The generalized quadratic covariation, an example} \label{sec6}
In this section, for $|b|<1$ we consider an example of Borel functions such that the generalized quadratic covariation exists. Recall that
$$
J_\varepsilon(f,t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
for all $\varepsilon>0$ and all Borel functions $f$, and the generalized quadratic covariation defined as follows
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}J_\varepsilon(f,t),
$$
provided the limit exists uniformly in probability (in short, ucp). For ucp-convergence we have the next result due to Russo {\em et al}~\cite{Russo2}.
\begin{lemma}[Russo {\em et al}~\cite{Russo2}]\label{lemm3.1-1}
Let $\{X^\varepsilon,\;\varepsilon>0\}$ be a family of continuous processes. We suppose
\begin{itemize}
\item For any $\varepsilon>0$, the process $t\mapsto X^\varepsilon_t$ is increasing;
\item There is a continuous process $X=(X_t,t\geq 0)$ such that $X^\varepsilon_t\to X_t$ in probability as $\varepsilon$ goes to zero.
\end{itemize}
Then $Z^\varepsilon$ converges to $X$ ucp.
\end{lemma}
\begin{proposition}\label{prop4.1}
Let $a>-1$, $|b|<1$ and $|b|<1+a$. Then we have
\begin{equation}\label{sec4-eq4.111111}
[B^{a,b},B^{a,b}]^{(a,b)}_t=\kappa_{a,b}t^{1+a+b},
\end{equation}
where
$$
\kappa_{a,b}=\frac1{(1+b){\mathbb B}(a+1,b+1)}.
$$
\end{proposition}
Let now $(X,Y)$ be a $2$-dimensional normal random variable with the density
$$
\varphi(x,y)=\frac1{2\pi}e^{-\frac1{2\rho^2} (\sigma_2^2x^2-2\mu xy+\sigma_1^2y^2)},
$$
where $E[X]=E[Y]=0,\sigma_1^2=E[X^2],\sigma_2^2=E[Y^2],\mu= E[XY]$ and $\rho^2=\sigma_1^2\sigma_2^2-\mu^2$. Then, an elementary calculus can show that
\begin{equation}\label{sec4-eq4.1-11}
E[X^2Y^2]=E[X^2]E[Y^2]+2(E[XY])^{2}.
\end{equation}
Denote by
\begin{align*}
h_s(\varepsilon):=E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2 -\kappa_{a,b}\varepsilon^{1+b}s^a
\end{align*}
for $\varepsilon\in (0,1)$ and $s>0$. By making substitution $u-s=+v\varepsilon$ we have
\begin{align*}
h_s(\varepsilon)&=\frac{1}{{\mathbb B}(a+1,b+1)}\int_s^{s+\varepsilon}u^a(s+\varepsilon-u)^bdu -\kappa_{a,b}\varepsilon^{1+b}s^a\\
&=\frac{\varepsilon^{1+b}}{{\mathbb B}(a+1,b+1)}\int_0^{1}(s+\varepsilon v)^a(1-v)^bdu-\kappa_{a,b}\varepsilon^{1+b}s^a\\
&=\varepsilon^{1+b}\kappa_{a,b}\left((1+b)\int_0^{1}(s+\varepsilon v)^a(1-v)^bdu-s^a\right)\\
&=\frac{\varepsilon^{1+b}}{{\mathbb B}(a+1,b+1)}\int_0^{1}\left\{(s+\varepsilon v)^a-s^a\right\}(1-v)^bdu.
\end{align*}
Notice that
$$
\left|(s+\varepsilon v)^a-s^a\right|=(s+\varepsilon v)^a-s^a\leq
(\varepsilon v)^a
$$
for all $0<a\leq 1,\;\varepsilon>0,\;0\leq v\leq 1$, and
$$
\left|(s+\varepsilon v)^a-s^a\right|=(s+\varepsilon v)^a-s^a\leq
C_a\varepsilon v(s+\varepsilon v)^{a-1}\leq C_a\varepsilon v(s+\varepsilon)^{a-1}
$$
for all $a>1,\;\varepsilon>0,\;0\leq v\leq 1$, and
\begin{align*}
\left|(s+\varepsilon v)^a-s^a\right|&=s^a-(s+\varepsilon v)^a
=s^{a}\left(1-(\frac{s}{s+\varepsilon v})^{-a}\right)\\
&\leq s^{a}\left(1-\frac{s}{s+\varepsilon v}\right)=s^{a}\frac{\varepsilon v}{s+\varepsilon v}
\leq s^a\frac{\varepsilon v}{s^\nu(\varepsilon v)^{1-\nu}}
=(\varepsilon v)^{\nu}s^{a-\nu}
\end{align*}
for all $-1<a<0,\;\varepsilon>0,\;0\leq v\leq 1$ and all $0<\nu<1+a$ by Young's inequality. We get
\begin{equation}\label{sec6-eq6.6}
h_s(\varepsilon)\leq
\begin{cases}
C_{a,b}(s+\varepsilon)^{a-1}\varepsilon^{2+b},& {\text { if $a>1$}},\\
C_{a,b}\varepsilon^{1+b+a},& {\text { if $0<a\leq 1$}},\\
C_{a,b}s^{a-\nu}\varepsilon^{1+b+\nu},& {\text { if $-1<a<0$ \;($0<\nu<1+a$)}}
\end{cases}
\end{equation}
for all $s>0$ and $\varepsilon>0$.
\begin{proof}[Proof of Proposition~\ref{prop4.1}]
Denote
$$
X_\varepsilon(t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
$$
for all $0<\varepsilon<1$. Then, it is sufficient to show that
$$
E\left(X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\right)^2\longrightarrow 0,
$$
as $\varepsilon$ tends to zero. We have
\begin{align*}
X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b} &=\frac{1+a+b}{\varepsilon^{1+b}}\left(\int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds -\kappa_{a,b}\frac{\varepsilon^{1+b}t^{1+a+b}}{1+a+b}\right)\\
&=\frac{1+a+b}{\varepsilon^{1+b}}\left(\int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds -\kappa_{a,b}\varepsilon^{1+b} \int_\varepsilon^{t+\varepsilon}s^{a+b}ds\right)\\
&\qquad+\kappa_{a,b}\left( (1+a+b) \int_\varepsilon^{t+\varepsilon}s^{a+b}ds- t^{1+a+b}\right)\\
&\equiv \Xi_1(a,b,\varepsilon,t)+\Xi_2(a,b,\varepsilon,t).
\end{align*}
We deduce from above
\begin{equation}\label{sec6-eq6.7}
E\left|X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\right|^2
\leq 2E|\Xi_1(a,b,\varepsilon,t)|^2+2|\Xi_2(a,b,\varepsilon,t)|^2
\end{equation}
for $t\geq 0$ and $\varepsilon>0$. Clearly, we have
\begin{align}\notag
|\Xi_2(a,b,\varepsilon,t)|&=\kappa_{a,b}\left| (1+a+b)\int_\varepsilon^{t+\varepsilon}s^{a+b}ds-t^{1+a+b}\right|\\
\label{sec6-eq6.8}
&=\kappa_{a,b}\left|(t+\varepsilon)^{1+a+b}-\varepsilon^{1+a+b} -t^{1+a+b}\right|=O(\varepsilon^{1\wedge (1+a+b)})
\end{align}
for each $t\geq 0$.
Now, let us estimate $E|\Xi_1(a,b,\varepsilon,t)|^2$. We have
\begin{equation}\label{sec6-eq6.9}
\begin{split}
E|\Xi_1(a,b,\varepsilon,t)|^2&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} E\left(\int_\varepsilon^{t+\varepsilon}
\left\{(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2-\kappa_{a,b} \varepsilon^{1+b}s^{a}\right\}s^{b}ds\right)^2\\
&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}A_\varepsilon(s,r)(sr)^{b}dsdr
\end{split}
\end{equation}
for each $t\geq 0$, where
\begin{align*}
A_\varepsilon(s,r):&=E\left(
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2-\kappa_{a,b} \varepsilon^{1+b}s^a\right)\left(
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2-\kappa_{a,b} \varepsilon^{1+b}r^a\right)\\
&=E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2(B^{a,b}_{r+\varepsilon} -B^{a,b}_r)^2+(\kappa_{a,b})^2\varepsilon^{2+2b}(sr)^a\\
&\qquad-\kappa_{a,b}\varepsilon^{1+b}
E\left((B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2s^a +(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2r^a\right).
\end{align*}
For all $r,s\geq 0$ and $\varepsilon>0$ by decomposing
\begin{align*}
E(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2 =h_s(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a
\end{align*}
we have
\begin{align*}
E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2(B^{a,b}_{r+\varepsilon} -&B^{a,b}_r)^2]=
E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2\right]
E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2\right]\\
&\hspace{2cm}+2\left(E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]\right)^{2}\\
&=\left[h_s(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a\right]
\left[h_r(\varepsilon)+\kappa_{a,b}\varepsilon^{1+b}s^a \right]+2(\mu_{s,r})^2
\end{align*}
by~\eqref{sec4-eq4.1-11}, where $\mu_{s,r}:=E\left[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]$, which yields
\begin{equation}\label{sec6-eq6.10}
A_\varepsilon(s,r)=h_s(\varepsilon)h_r(\varepsilon)+2(\mu_{s,r})^2.
\end{equation}
On the other hand, for $0<s-r<\varepsilon\leq 1$ we have
\begin{align*}
(\mu_{s,r})^2&\leq E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2]E[(B^{a,b}_{r+\varepsilon} -B^{a,b}_{r})^2]\\
&\leq (s+\varepsilon)^a(r+\varepsilon)^a\varepsilon^{2+2b}
\leq (s+\varepsilon)^a(r+\varepsilon)^a\frac{\varepsilon^{2+2b+\gamma} }{(s-r)^\gamma}
\end{align*}
for all $0\leq \gamma\leq 1$. It follows from Lemma~\ref{lem3.3} that
\begin{equation}\label{sec6-eq6.11}
\begin{split}
(\mu_{s,r})^2&\leq C_{a,b}\left(r^a\vee s^a\right)^{2\alpha} [(s+\varepsilon)(r+\varepsilon)]^{a(1-\alpha)} \frac{\varepsilon^{2+2b+2\alpha(1-b)} }{(s-r)^{2(1+b)\alpha}}1_{\{s-r>\varepsilon\}}\\
&\qquad+(s+\varepsilon)^a(r+\varepsilon)^a \frac{\varepsilon^{2+2b+\alpha} }{(s-r)^\alpha}1_{\{s-r\leq \varepsilon\}}
\end{split}
\end{equation}
for all $0<\alpha<\frac1{2(1+b)}\wedge 1$. Combining this with~\eqref{sec6-eq6.6},~\eqref{sec6-eq6.7},~\eqref{sec6-eq6.8}, ~\eqref{sec6-eq6.9} and~\eqref{sec6-eq6.10} we see that
\begin{align*}
E|\Xi_1(a,b,\varepsilon,t)|^2&=\frac{(1+a+b)^2}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}A_\varepsilon(s,r)(sr)^{b}drds \longrightarrow 0
\end{align*}
for all $t\geq 0$, as $\varepsilon$ tends to zero, and the proposition follows.
\end{proof}
Recall that the local H\"{o}lder index $\gamma_0$
of a continuous paths process $\{X_t: t\geq 0\}$ is the supremum of
the exponents $\gamma$ verifying, for any $T>0$:
$$
P(\{\omega: \exists L(\omega)>0, \forall s,t \in[0,T],
|X_t(\omega)-X_s(\omega)|\leq L(\omega)|t-s|^\gamma\})=1.
$$
The next lemma is considered by Gradinaru-Nourdin~\cite{Grad3}.
\begin{lemma}\label{Grad-Nourdin}
Let $g:{\mathbb R}\to {\mathbb R}$ be a continuous function satisfying
\begin{equation}\label{eq4.2-Gradinaru--Nourdin}
|g(x)-g(y)|\leq C|x-y|^a(1+x^2+y^2)^b,\quad (C>0,0<a\leq 1,b>0),
\end{equation}
for all $x,y\in {\mathbb R}$ and let $X$ be a locally H\"older
continuous paths process with index $\gamma\in (0,1)$. Assume that
$V$ is a bounded variation continuous paths process. Set
$$
X^{g}_\varepsilon(t)=\int_\varepsilon^{t+\varepsilon} g(\frac{X_{s+\varepsilon}-X_s
}{\varepsilon^\gamma})ds
$$
for $t\geq 0$, $\varepsilon>0$. If for each $t\geq 0$, as
$\varepsilon\to 0$,
\begin{equation}\label{condition}
\|X^{g}_\varepsilon(t)-V_t\|_{L^2}^2=O(\varepsilon^\alpha)
\end{equation}
with $\alpha>0$, then, for any $t\geq 0$, $\lim_{\varepsilon\to
0}X^{g}_\varepsilon(t)=V_t$ almost surely, and if $g$ is
non-negative, for any continuous stochastic process $\{Y_t:\;t\geq
0\}$,
\begin{equation}
\lim_{\varepsilon\to 0}
\int_\varepsilon^{t+\varepsilon} Y_sg(\frac{X_{s+\varepsilon}-X_s}{\varepsilon^\gamma})ds
\longrightarrow \int_0^tY_sdV_s,
\end{equation}
almost surely, uniformly in $t$ on each compact interval.
\end{lemma}
Recall that
$$
X_\varepsilon(t)=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
$$
for all $0<\varepsilon<1$. The proof of Proposition~\ref{prop4.1} points out that there exists $\beta\in (0,1)$ such that
\begin{align*}
\|X_\varepsilon(t)-\kappa_{a,b}t^{1+a+b}\|_{L^2}=O(\varepsilon^{\beta}), \qquad \varepsilon\to 0
\end{align*}
for all $t\geq 0$. Notice that $g(x)=x^2$ satisfies the condition~\eqref{eq4.2-Gradinaru--Nourdin}. We obtain the convergence
\begin{align*}
\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2s^{b}ds
\longrightarrow \kappa_{a,b}\int_0^tf(B^{a,b}_s)ds^{1+a+b},
\end{align*}
almost surely, uniformly in $t$ on each compact interval by taking $Y_s=f(B^{a,b}_s)s^{b}$ for $s\geq 0$. Clearly, by the H\"older continuity of weighted-fBm $B^{a,b}$, we get
\begin{align*}
\frac{1+a+b}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon}o(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds\longrightarrow 0
\end{align*}
in $L^1(\Omega)$, as $\varepsilon \to 0$. Thus, we have obtained the next result.
\begin{corollary}
Let $a>-1$, $|b|<1$, $|b|<1+a$ and let $f\in
C^1({\mathbb R})$. Then the generalized quadratic covariation $[f(B^{a,b}),B^{a,b}]^{(a,b)}$ exists and
\begin{align}\label{sec6-eq6.2}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t&=\kappa_{a,b} \int_0^tf'(B^{a,b}_s)ds^{1+a+b}.
\end{align}
\end{corollary}
\section{The generalized quadratic covariation, existence}~\label{sec5}
In this section, for $-1<b<0$ we shall study the existence of the
{\it generalized quadratic covariation} more general class of
functions than the one of class $C^1$. Recall that
$$
J_\varepsilon(f,t):=\frac{1+a+b}{\varepsilon^{1+b}} \int_\varepsilon^{t+\varepsilon}\left\{ f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds
$$
for all $\varepsilon>0$ and all Borel functions $f$. Consider the decomposition
\begin{equation}\label{sec4-eq4.000000}
\begin{split}
\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} &\left\{f(B^{a,b}_{
s+\varepsilon})-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)s^{b}ds\\
&=\frac{1}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_{
s+\varepsilon})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\
&\qquad\qquad\qquad-\frac{1}{\varepsilon^{1+b}}
\int_\varepsilon^{t+\varepsilon} f(B^{a,b}_s)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^bds\\
&\equiv I_\varepsilon^{+}(f,t)-I_\varepsilon^{-}(f,t),
\end{split}
\end{equation}
and define the set ${\mathscr H}$ of measurable functions $f$ on
${\mathbb R}$ such that $\|f\|_{{\mathscr H}}<\infty$, where
\begin{align*}
\|f\|_{{\mathscr H}}:=&\sqrt{\int_0^{T+1}\int_{\mathbb
R}|f(x)|^2e^{-\frac{x^2}{2s^{1+a+b}
}}\frac{dxds}{\sqrt{2\pi}s^{(1-a-b)/2}}}.
\end{align*}
For the set ${\mathscr H}$ we have
\begin{itemize}
\item [(1)] ${\mathscr H}$ is a Banach space.
\item [(2)] ${\mathscr H}\supset C^\infty_0({\mathbb R})$, the set of infinitely differentiable functions with compact support.
\item [(3)] the set ${\mathscr E}$ of elementary functions of the form
$$
f_\triangle(x)=\sum_{i}f_{i}1_{(x_{i-1},x_{i}]}(x)
$$
is dense in ${\mathscr H}$, where $\{x_i,0\leq i\leq l\}$ is an
finite sequence of real numbers such that $x_i<x_{i+1}$.
\item [(4)] the space ${\mathscr H}$ contains all Borel functions $f$ satisfying
$$
|f(x)|\leq Ce^{\beta x^2},\qquad x\in {\mathbb R}
$$
with $\beta<\frac14T^{-(1+a+b)}$.
\end{itemize}
For simplicity we let $T=1$ in the following discussions. The main result of this section is to explain and prove the
following theorem.
\begin{theorem}\label{th6.2}
Let $a>-1$, $-1<b<0$, $|b|<1+a$ and $f\in {\mathscr H}$. Then the generalized quadratic covariation $[f(B^{a,b}),B^{a,b}]^{(a,b)}$ exists, and we have
\begin{align}\label{th4.1-eq}
E\left|[f(B^{a,b}),B^{a,b}]^{(a,b)}_t\right|^2\leq C_{a,b,T}\|f\|_{{\mathscr H}}^2
\end{align}
for all $a\geq 0,t\in [0,1]$.
\end{theorem}
For simplicity we let $T=1$ in the rest of this section.
\begin{lemma}\label{lem6.2}
Let $a>-1,\;-1<b<0,\;|b|<1+a$ and let $f\in {\mathscr H}$. We then have
\begin{align}
&E\left|I_\varepsilon^{-}(f,t)\right|^2\leq C_{a,b}\|f\|_{{\mathscr H}}^2,\\
&E\left|I_\varepsilon^{+}(f,t)\right|^2\leq C_{a,b}\|f\|_{{\mathscr H}}^2
\end{align}
for all $0<\varepsilon<1$ and $t\in [0,1]$.
\end{lemma}
\begin{proof}
We prove only the first estimate, and similarly one can prove the second estimate. By the denseness of ${\mathscr E}$ in ${\mathscr H}$ we only need to show that the lemma holds for all $f\in {\mathscr E}$. Consider the function $\zeta$ on ${\mathbb R}$ by
\begin{equation}
\zeta(x):=
\begin{cases}
ce^{\frac1{(x-1)^2-1}}, &{\text { $x\in (0,2)$}},\\
0, &{\text { otherwise}},
\end{cases}
\end{equation}
where $c$ is a normalizing constant such that $\int_{\mathbb
R}\zeta(x)dx=1$, and define the mollifiers
\begin{equation}\label{sec7-eq7.4}
\zeta_n(x):=n\zeta(nx),\qquad n=1,2,\ldots.
\end{equation}
For $f_\triangle\in {\mathscr E}$ consider the sequence of smooth functions
$$
f_{\triangle,n}(x):=\int_{\mathbb R}f_{\triangle}(x-{y})\zeta_n(y)dy=\int_0^2 f_{\triangle}(x-\frac{y}n)\zeta(y)dy,\quad x\in {\mathbb R}.
$$
Then $f_{\triangle,n}\in C_0^\infty({\mathbb R})$ for all $n\geq 1$ and $f_{\triangle,n}\to f_{\triangle}$ in ${\mathscr H}$, as $n$ tends to infinity. Thus, by approximating we may assume that $f$ is an infinitely differentiable function with compact support. It follows from the duality relationship that
\begin{equation}\label{sec4-eq4.800}
\begin{split}
E[f(&B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]\\
&=E\left[f(B^{a,b}_{s})f(B^{a,b}_{
r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \int_r^{r+\varepsilon}dB^{a,b}_l\right]\\
&=E\left\langle D^{a,b}\left(
f(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right), 1_{[r,r+\varepsilon]}\right\rangle_{{\mathcal H}_{a,b}}\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]E\left[
f'(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]E\left[
f(B^{a,b}_{s})f'(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) \right]\\
&\qquad
+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s) \right]E\left[
f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad
+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\equiv \sum_{i=1}^5 \Psi_i(s,r,\varepsilon).
\end{split}
\end{equation}
In order to end the proof we claim now that
\begin{equation}\label{eq4.4}
\frac{1}{\varepsilon^{2+2b}}\left| \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} \Psi_i(s,r,\varepsilon)(sr)^{b}dsdr \right|\leq C_{a,b}\|f\|^2_{{\mathscr H}},\qquad i=1,2,\ldots,5
\end{equation}
for all $\varepsilon>0$ small enough.
{\bf For $i=5$} we have
\begin{align*}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|&\leq \sqrt{E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)^2E(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)^2]}\\
&\leq C_{a,b}[(r+\varepsilon)(s+\varepsilon)]^{a/2} \varepsilon^{1+b}\\
&\leq C_{a,b}[(r+\varepsilon)(s+\varepsilon)]^{a/2}\frac{ \varepsilon^{2+2b}}{(s-r)^{1+b}}
\end{align*}
for $0<s-r<\varepsilon\leq 1$, which gives
\begin{align*}
I_\varepsilon:&= \frac{1}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_{s-\varepsilon}^s
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|E[f^2(B^{a,b}_{s})](sr)^{b}
drds\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}\int_{s-\varepsilon}^s \frac{(r^a\vee s^a)(sr)^{b}}{(s-r)^{1+b}} E[f^2(B^{a,b}_{s})]drds\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}
s^bEf^2(B^{a,b}_{s})ds
\int_0^s \frac{r^{b}(r^a\vee s^a)}{(s-r)^{1+b}} dr\\
&=C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{b+a}Ef^2(B^{a,b}_{s})ds \leq C_{a,b}\int_0^{2}s^{a+b}Ef^2(B^{a,b}_{s})ds
\end{align*}
for $0<s-r<\varepsilon\leq 1$. Moreover, by~\eqref{sec3-eq3.8==} with $\beta=1$ and $\nu=-b$ we have
\begin{align*}
II_\varepsilon:&=\frac{1}{\varepsilon^{2+2b}} \int_{\varepsilon}^{t+\varepsilon} \int_\varepsilon^{s-\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|E[f^2(B^{a,b}_{s})] (sr)^{b}drds \\
&\leq C_{a,b}\int_{\varepsilon}^{t+\varepsilon}
s^bE[f^2(B^{a,b}_{s})]ds\int_\varepsilon^{s-\varepsilon}(r^a\vee s^a)\frac{r^bdr}{(s-r)^{1+b}} \\
&\leq C_{a,b} \int_{\varepsilon}^{t+\varepsilon}s^b Ef^2(B^{a,b}_{s})ds \int_0^s
\frac{r^{b}(r^a\vee s^a)}{(s-r)^{1+b}}dr\\
&\leq C_{a,b}\int_{\varepsilon}^{t+\varepsilon}s^{a+b} Ef^2(B^{a,b}_{s})ds\leq C_{a,b}\int_{0}^{2}s^{a+b} Ef^2(B^{a,b}_{s})ds
\end{align*}
for all $0<\varepsilon\leq 1$. It follows that
\begin{align*}
& \frac{1}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} |\Psi_5(s,r,\varepsilon)|(sr)^{b} drds \\
&\leq \frac{1 }{2\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|[Ef^2(B^{a,b}_{s})+Ef^2(B^{a,b}_{r})] (sr)^{b}drds\\
&\leq \frac{1}{\varepsilon^{2+2b}}\int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{t+\varepsilon}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|Ef^2(B^{a,b}_{s})(sr)^{b}drds\\
&\leq \frac{2}{\varepsilon^{2+2b}}\int_\varepsilon^{t+\varepsilon} \int_\varepsilon^{s}
|E[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon} -B^{a,b}_s)]|Ef^2(B^{a,b}_{s})(sr)^{b}drds\\
&\leq 2(I_\varepsilon+II_\varepsilon)\leq C_{a,b}
\int_{0}^{2}s^{a+b} E[f^2(B^{a,b}_{s})]ds
\end{align*}
for all $0<\varepsilon\leq 1$.
{\bf For $i=1$}, by Lemma~\ref{lem3.4} and Lemma~\ref{lem3.5} we have
\begin{align*}
\frac{1}{\varepsilon^{2+2b}}& \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon} |\Psi_1(s,r,\varepsilon)|(sr)^{b}drds\\
&\leq \frac{C_{a,b}}{2\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]
E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(Ef^2(B^{a,b}_s)
+Ef^2(B^{a,b}_r)\right)(sr)^{b}dsdr\\
&= \frac{C_{a,b}}{\varepsilon^{2+2b}} \int_\varepsilon^{t+\varepsilon}\int_\varepsilon^{t+\varepsilon}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]
E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right) Ef^2(B^{a,b}_s)(sr)^{b}dsdr\\
&\leq C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{a+b}ds\int_\varepsilon^{s}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{a+b}dr\\
&\qquad +C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{a+b}ds \int_s^{s+\varepsilon}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{a+b}dr\\
&\qquad +C_{a,b}\int_\varepsilon^{t+\varepsilon}s^{2a+b}ds \int_{s+\varepsilon}^{t+\varepsilon}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)Ef^2(B^{a,b}_s)r^{b}dr\\
&\leq C_{a,b}
\int_{0}^{2}s^{a+b} E[f^2(B^{a,b}_{s})]ds
\end{align*}
for all $0<\varepsilon\leq 1$. Similarly, we can obtain the estimate~\eqref{eq4.4} for $i=2,3,4$, and
the lemma follows.
\end{proof}
\begin{lemma}\label{lem6.3}
Let $a>-1,\;-1<b<0,\;|b|<1+a$ and let $f$ be an infinitely differentiable function with compact support. We then have
\begin{align}
\Pi_1(t,\varepsilon_i,\varepsilon_j):= \frac1{\varepsilon_i^{2+2b}} E\Bigl|\int_{t+\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_i}-B^{a,b}_s)s^bds\Bigr|^2\longrightarrow 0
\end{align}
for all $0<\varepsilon_2<\varepsilon_1<1$ and $t\in [0,1]$, as $\varepsilon_1\to 0$, where $i,j=1,2$ and $i\neq j$.
\end{lemma}
\begin{proof}
From the proof of Lemma~\ref{lem6.2} one can easily prove the result.
\end{proof}
Now we can show our main result.
\begin{proof}[Proof of Theorem~\ref{th6.2}]
From Lemma~\ref{lem6.2}, it is enough to show that $I_{\varepsilon_1}^{-}(f,t)$ and $I_{\varepsilon_1}^{+}(f,t)$ are two Cauchy's sequences in $L^2(\Omega)$ for all $t\in [0,1]$. That is,
\begin{equation}\label{sec40-eq3-1}
E\left|I_{\varepsilon_1}^{-}(f,t)-I_{\varepsilon_2}^{-}(f,t)\right|^2
\longrightarrow 0,
\end{equation}
and
\begin{equation}\label{sec40-eq3-2}
E\left|I_{\varepsilon_1}^{+}(f,t)-I_{\varepsilon_2}^{+}(f,t)\right|^2
\longrightarrow 0
\end{equation}
for all $t\in [0,1]$, as $\varepsilon_1,\varepsilon_2\downarrow 0$. We prove only the convergence~\eqref{sec40-eq3-1}, and similarly one can prove~\eqref{sec40-eq3-2}. Without loss of generality one may assume that $\varepsilon_1>\varepsilon_2$. It follows that
\begin{align*}
&E\bigl|I_{\varepsilon_1}^{-}(f,t)-I_{\varepsilon_2}^{-}(f,t)\bigr|^2
\\
&\leq 3E\Bigl|\frac1{\varepsilon_1^{1+b}} \int_{t+\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)s^bds\Bigr|^2\\
&\qquad+3E\Bigl|\frac1{\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s)s^bds\Bigr|^2\\
&+3E\Bigl|\frac1{\varepsilon_1^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)s^bds- \frac1{\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1}f(B^{a,b}_s) (B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s)s^bds\Bigr|^2\\
&\equiv \Pi_1(t,\varepsilon_1,\varepsilon_2) +\Pi_1(0,\varepsilon_2,\varepsilon_1) +\Pi_2(t,\varepsilon_1,\varepsilon_2).
\end{align*}
In order to end the proof, it is enough to check $\Pi_2(t,\varepsilon_1,\varepsilon_2)\to 0$, as $\varepsilon_1,\varepsilon_2\to 0$. We have
\begin{align*}
\Pi_2&(t,\varepsilon_1,\varepsilon_2)\\
&=\frac1{\varepsilon_1^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s) (B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(sr)^bdrds\\
&\quad-
\frac2{\varepsilon_1^{1+b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)(sr)^bdrds\\
&\quad+\frac1{\varepsilon_2^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} Ef(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_2}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)(sr)^bdrds\\
&\equiv \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}\left\{\varepsilon_2^{1+b} \Phi_{s,r}(1,\varepsilon_1)-\varepsilon_1^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds\\
&\quad+
\frac1{\varepsilon_1^{1+b}\varepsilon_2^{2+2b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \left\{\varepsilon_1^{1+b}\Phi_{s,r}(1,\varepsilon_2)
-\varepsilon_2^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds,
\end{align*}
where
$$
\Phi_{s,r}(1,\varepsilon) =E\left[f(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right],
$$
and
$$
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2) =E\left[f(B^{a,b}_s)f(B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right].
$$
In order to estimate $\Phi_{s,r}(1,\varepsilon)$ and $\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)$, by approximating we may assume that $f$ is an infinitely differentiable function with compact support. We then have by~\eqref{sec4-eq4.800},
\begin{align*}
\Phi_{s,r}(1,\varepsilon)&=\sum_{i=1}^5\Psi_i(s,r,{\varepsilon})\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right] E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad\qquad+E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)
(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]E\left[
f(B^{a,b}_{s})f(B^{a,b}_{r})\right]
\end{align*}
and
\begin{align*}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)&=
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]E\left[
f'(B^{a,b}_{s})f(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) \right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]E\left[
f(B^{a,b}_{s})f'(B^{a,b}_{r})(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) \right]\\
&\qquad\qquad +E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)
\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&=E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&+E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad +E\left[B_{r}(B^{a,b}_{r+\varepsilon_2}-B_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B_s)\right]
E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad +E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]
E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]\\
&\qquad\qquad +E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)
\right]E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right].
\end{align*}
Denote
\begin{align*}
A_1(s,r,\varepsilon,j):&=\varepsilon_j^{1+b} E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_{11}(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right] E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_{12}(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b}
E\left[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_3(s,r,\varepsilon,j):&=\varepsilon_j^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b} E\left[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
E\left[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)\right]\\
A_4(s,r,\varepsilon,j):&=\varepsilon_j^{1+b}
E\left[(B^{a,b}_{r+\varepsilon}-B^{a,b}_r) (B^{a,b}_{s+\varepsilon}-B^{a,b}_s)\right]\\
&\qquad-\varepsilon^{1+b}
E\left[(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s) (B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)\right]
\end{align*}
with $j=1,2$. It follows that
\begin{align*}
\varepsilon_j^{1+b}&\Phi_{s,r}(1,\varepsilon_i) -\varepsilon_i^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\\
&=A_1(s,r,\varepsilon_i,j)E\left[f''(B^{a,b}_{s})f(B^{a,b}_{r})\right]\\
&\qquad
+(A_{11}(s,r,\varepsilon_i,j)+A_{12}(s,r,\varepsilon_i,j)) E\left[f'(B^{a,b}_{s})f'(B^{a,b}_{r})\right]\\
&\qquad+A_3(s,r,\varepsilon_i,j)E\left[f(B^{a,b}_{s})f''(B^{a,b}_{r})\right]
+A_4(s,r,\varepsilon_i,j)E\left[f(B^{a,b}_{s})f(B^{a,b}_{r})\right]
\end{align*}
with $i,j\in\{1,2\}$ and $i\neq j$. In order to end the proof we claim
that the following convergence
\begin{align}\label{sec4-Con-eq1}
\frac1{\varepsilon_i^{2+2b}\varepsilon_j^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \left\{\varepsilon_j^{1+b}\Phi_{s,r}(1,\varepsilon_i) -\varepsilon_i^{1+b}
\Phi_{s,r}(2,\varepsilon_1,\varepsilon_2)\right\}(sr)^bdrds \longrightarrow 0
\end{align}
with $i,j\in\{1,2\}$ and $i\neq j$, as $\varepsilon_1,\varepsilon_2\to 0$. By symmetry, we only need to show that this holds for $i=1,j=2$. This will be done in three parts.
{\bf Part I}. The following convergence holds:
\begin{align}\label{step1-eq1}
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\frac{A_4(s,r,\varepsilon_1,2)} {\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} E[f(B^{a,b}_{s})f(B^{a,b}_{r})](sr)^bdrds \longrightarrow 0,
\end{align}
as $\varepsilon_1,\varepsilon_2\to 0$. Clearly, we have
\begin{align*}
|E[(B^{a,b}_{r+\varepsilon_i}-B^{a,b}_r)&(B^{a,b}_{s+\varepsilon_j} -B^{a,b}_s)]|\leq \sqrt{E[(B^{a,b}_{r+\varepsilon_i}-B^{a,b}_r)^2 E(B^{a,b}_{s+\varepsilon_j} -B^{a,b}_s)^2]}\\
&\leq C_{a,b}[(r+\varepsilon_i)(s+\varepsilon_j)]^{a/2} \varepsilon_i^{(1+b)/2}\varepsilon_j^{(1+b)/2}\\
&\leq C_{a,b}[(r+\varepsilon_i)(s+\varepsilon_j)]^{a/2} \frac{\varepsilon_i^{1+b+\gamma} \varepsilon_j^{1+b}
}{(s-r)^{1+b+\gamma}}
\end{align*}
for $0<s-r<\varepsilon_i\wedge \varepsilon_j\leq 1$ and $0<\gamma<-b$, where $i,j\in \{1,2\}$. Combining this with~\eqref{sec3-eq3.8==} (taking $\beta=1$), we get
\begin{align*}
|E[(&B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)]|\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \left(\frac{\varepsilon_1^{2+b-\nu}}{(s-r)^{1-\nu}}
1_{\{s-r>\varepsilon_1\}}+ \frac{\varepsilon_1^{2+2b+\gamma}}{(s-r)^{1+b+\gamma}}
1_{\{0<s-r\leq \varepsilon_1\}}\right)\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \frac{\varepsilon_1^{2+b-\nu}}{(s-r)^{1-\nu}}
\end{align*}
and
\begin{align*}
|E[(&B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right)\left( \frac{\varepsilon_1^{1+b-\nu}\varepsilon_2}{(s-r)^{1-\nu}}
1_{\{s-r>\varepsilon_2\}}+ \frac{\varepsilon_1^{1+b+\gamma}\varepsilon_2^{1+b}}{(s-r)^{1+b+\gamma}}
1_{\{0<s-r\leq \varepsilon_2\}}\right)\\
&\leq C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right) \frac{\varepsilon_1^{1-\nu}\varepsilon_2^{1+b}}{(s-r)^{1-\nu}}
\end{align*}
for all $s>r>0$ and $0<\nu<(1+b)\wedge (-b)$ by taking $b+\gamma+\nu\geq 0$. It follows that
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |A_4(s,r,\varepsilon_1,2)|&\leq \frac{C_{a,b}\left(r^a\vee (s+\varepsilon_1)^a\right)}{(s-r)^{1-\nu}} \varepsilon_1^{-b-\nu}\longrightarrow 0
\end{align*}
for all $s>r>0$ and $\nu<-b$, as $\varepsilon_1,\varepsilon_2\to 0$.
On the other hand, from the above proof we have also
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |A_4(s,r,\varepsilon_1,2)|&\leq \frac1{\varepsilon_1^{2+2b}} |E[(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)]|\\
&\qquad+\frac1{\varepsilon_1^{1+b}\varepsilon_2^{1+b}}
|E[(B^{a,b}_{s+\varepsilon_1} -B^{a,b}_s)(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\\
&\leq C_{a,b}\frac{r^a\vee (s+\varepsilon_1)^a}{(s-r)^{1+b}}
\end{align*}
for all $s>r>0$ and $\varepsilon_1,\varepsilon_2>0$, and
$$
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} \frac{r^a}{(s-r)^{1+b}} |E[f(B^{a,b}_{s})f(B^{a,b}_{r})]|(sr)^bdrds\leq C_{a,b}\|f\|^2_{\mathscr H}
$$
for any $0<\varepsilon_1,\varepsilon_2<1$. Using Lebesgue's dominated convergence theorem we can deduce the convergence~\eqref{step1-eq1}.
{\bf Part II}. The following convergence holds:
\begin{align}\label{step2-eq1}
\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\frac{A_{11}(s,r,\varepsilon_1,2)+A_{12}(s,r,\varepsilon_1,2)}{ \varepsilon_1^{2+2b}\varepsilon_2^{1+b}} E[f'(B^{a,b}_{s})f'(B^{a,b}_{r})](sr)^bdrds\to 0,
\end{align}
as $\varepsilon_1,\varepsilon_2\to 0$. By Lemma~\ref{lem3.4} we have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}& \left(|A_{11} (s,r,\varepsilon_1,2)|+|A_{12}(s,r,\varepsilon_1,2)|\right)\\
&\leq |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot\left(\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]| +\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\quad +|E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\qquad\cdot\left(
\varepsilon_2^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]| +\varepsilon_1^{1+b}
|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}s^ar^a
\end{align*}
for all $s>r>0$, and moreover, by Lemma~\ref{lem3.1} and Lemma~\ref{lem3.5} we have
\begin{align*}
&\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
s^ar^a|E[f'(B^{a,b}_{s})f'(B^{a,b}_{r})]|(sr)^bdrds\\
&\;\;\leq C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)\left(E|f(B^{a,b}_s)|^2
+E|f(B^{a,b}_r)|^2\right)(sr)^{a+b}drds\\
&\;\;=C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1}
\left(\frac{(rs)^{\frac12(1+a+b)}}{\rho^2_{s,r}} +\frac{\mu_{s,r}}{\rho^2_{s,r}}\right)E|f(B^{a,b}_s)|^2(sr)^{a+b}drds \leq C_{a,b}\|f\|_{\mathscr H}^2
\end{align*}
for all $\varepsilon_1,\varepsilon_2>0$
On the other hand, we have
\begin{align*}
E[B^{a,b}_{r}&(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]\\
&=\frac1{2{\mathbb B}(1+a,1+b)}
\left(\int_0^ru^a(r+\varepsilon-u)^bdu-{\mathbb B}(1+a,1+b)r^{1+a+b}\right)\\
&=\frac1{2{\mathbb B}(1+a,1+b)} \int_0^{\frac{r}{r+\varepsilon}}x^a(1-x)^bdx \left\{(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right\}\\
&\qquad-\frac{r^{1+a+b}}{2{\mathbb B}(1+a,1+b)} \int_{\frac{r}{r+\varepsilon}}^1x^a(1-x)^bdx\\
&\equiv \frac1{2{\mathbb B}(1+a,1+b)} \left\{\Lambda_1(r,\varepsilon)-\Lambda_2(r,\varepsilon)\right\}
\end{align*}
for $r>0$ and $\varepsilon>0$, and
\begin{align*}
\Bigl|\varepsilon_2^{1+b}
E[&B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]
-\varepsilon_1^{1+b}E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2} -B^{a,b}_r)]\Bigr|\\
&\leq \varepsilon_2^{1+b}|\Lambda_1(r,\varepsilon_1)| +\varepsilon_1^{1+b}|\Lambda_1(r,\varepsilon_2)| +\left|\varepsilon_2^{1+b}\Lambda_2(r,\varepsilon_1) -\varepsilon_1^{1+b}\Lambda_2(r,\varepsilon_2)\right|
\end{align*}
for $r>0$ and $\varepsilon_1>\varepsilon_2>0$. Notice that for $r>0$ and $\varepsilon>0$,
\begin{align*}
|\Lambda_1(r,\varepsilon)|&= \int_0^{\frac{r}{r+\varepsilon}}x^a(1-x)^bdx \left|(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right|\\
&\leq C_{a,b}\left(\frac{r}{r+\varepsilon}\right)^{1+a} \left|(r+\varepsilon)^{1+a+b}- r^{1+a+b}\right|\\
&\leq C_{a,b}\left(\frac{r}{r+\varepsilon}\right)^{1+a}
(r+\varepsilon)^{a+b}\varepsilon\leq C_{a,b}r^{a+b}\varepsilon
\end{align*}
by~\eqref{lem3.4-eq7} and the fact
\begin{equation}\label{step2-eq2-1}
y^\alpha-x^\alpha\leq C_{\alpha}y^{\alpha-1}(y-x),\quad y>x>0,\quad \alpha>0,
\end{equation}
and
\begin{align*}
|\varepsilon_2^{1+b}&\Lambda_2(r,\varepsilon_1) -\varepsilon_1^{1+b}\Lambda_2(r,\varepsilon_2)|\\
&=C_{a,b}r^{1+a+b}\left|\varepsilon_2^{1+b} \int_{\frac{r}{r+\varepsilon_1}}^1x^a(1-x)^bdx-
\varepsilon_1^{1+b} \int_{\frac{r}{r+\varepsilon_2}}^1x^a(1-x)^bdx\right|\\
&=C_{a,b}r^{1+a+b}(\varepsilon_1\varepsilon_2)^{1+b} \left|\frac1{\varepsilon_1^{1+b}} \int_{\frac{r}{r+\varepsilon_1}}^1x^a(1-x)^bdx-
\frac1{\varepsilon_2^{1+b}} \int_{\frac{r}{r+\varepsilon_2}}^1x^a(1-x)^bdx\right|\\
&\equiv C_{a,b}r^{1+a+b}(\varepsilon_1\varepsilon_2)^{1+b}
\Lambda_3(r,\varepsilon_1,\varepsilon_2)
\end{align*}
for $r>0$ and $\varepsilon_1>\varepsilon_2>0$. We get
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}& |A_{11}(s,r,\varepsilon_1,2)|\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\cdot\left(\varepsilon_2^{1+b}
E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)] -\varepsilon_1^{1+b} E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]\right)\\
&\leq C_{a,b}s^ar^{a+b}\left(\varepsilon_1^{-b}+\varepsilon_2^{-b}+
r\Lambda_3(r,\varepsilon_1,\varepsilon_2)\right)\longrightarrow 0
\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s,r>0$ and $-1<b<0$ because
$$
\Lambda_3(r,\varepsilon_1,\varepsilon_2)\longrightarrow 0
$$
as $0<\varepsilon_2<\varepsilon_1\to 0$ by the convergence
$$
\lim_{x\uparrow 1}\frac1{(1-x)^{1+b}}\int_x^1u^a(1-u)^bdu=\frac1{1+b},
$$
and similarly we also have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}}&|A_{12} (s,r,\varepsilon_1,2)| \longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s,r>0$, and the convergence~\eqref{step2-eq1} follow from Lebesgue's dominated convergence theorem again.
{\bf Part III}. The following convergence holds:
\begin{equation}\label{step3-eq1}
\begin{split}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} \int_{\varepsilon_2}^{t+\varepsilon_1} &\int_{\varepsilon_2}^{t+\varepsilon_1} \Bigl(|A_1(s,r,\varepsilon_1,2)E[f''(B^{a,b}_{s})f(B^{a,b}_{r})]|\\
&\qquad +|A_3(s,r,\varepsilon_1,2)E[f(B^{a,b}_{s})f''(B^{a,b}_{r})]|
\Bigr)(sr)^bdrds \longrightarrow 0,
\end{split}
\end{equation}
as $\varepsilon_1,\varepsilon_2\to 0$. Clearly, for all $s>r>0$ and all $\varepsilon_1,\varepsilon_2>0$ we have
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} (|&A_1(s,r,\varepsilon_1,2)|+|A_3(s,r,\varepsilon_1,2)|)
\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad\cdot\left(\varepsilon_2^{1+b}| E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|+ \varepsilon_1^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)] |\right)\\
&+\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(
\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|
+\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}s^ar^a
\end{align*}
by Lemma~\ref{lem3.4}, and
\begin{align*}
\int_{\varepsilon_2}^{t+\varepsilon_1} &\int_{\varepsilon_2}^{t+\varepsilon_1} s^ar^a\Bigl(|E[f''(B^{a,b}_{s})f(B^{a,b}_{r})]| +|E[f(B^{a,b}_{s})f''(B^{a,b}_{r})]|
\Bigr)(sr)^bdrds\\
&\leq C_{a,b}\int_{\varepsilon_2}^{t+\varepsilon_1} \int_{\varepsilon_2}^{t+\varepsilon_1} s^ar^a\frac{r^{1+a+b}+s^{1+a+b}}{\rho^2_{s,r}}( E|f(B^{a,b}_s)|^2+E|f(B^{a,b}_r)|^2)(sr)^bdrds\\
&\leq C_{a,b} \int_{\varepsilon_2}^{t+\varepsilon_1} s^{a+b}E|f(B^{a,b}_s)|^2ds\leq C_{a,b}\|f\|_{{\mathscr H}}^2
\end{align*}
by Lemma~\ref{lem3.5} and Lemma~\ref{lem3.1}.
On the other hand, by the fact~\eqref{step2-eq2-1} we have
\begin{equation}\label{sec4-eq4.2200-2}
\begin{split}
\int_r^{r+\varepsilon}u^a(s-u)^bdu&\leq \frac1{1+b}\left(r^a\vee (r+\varepsilon)^a\right) \left((s-r)^{1+b}-(s-r-\varepsilon)^{1+b}\right)\\
&\leq \frac1{1+b}\left(r^a\vee s^a\right)(s-r)^{1+b-\beta}\varepsilon^\beta
\end{split}
\end{equation}
for $r+\varepsilon<s$ and $1+b<\beta<1$, which implies that
\begin{equation}\label{sec4-eq4.2200}
\begin{split}
|E[B^{a,b}_s&(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]| 1_{\{s-r>\varepsilon\}}=C_{a,b}\left(\int_0^{r+\varepsilon}u^a[(s-u)^b +({r+\varepsilon}-u)^b]du\right.\\
&\hspace{3cm}\left.-
\int_0^ru^a[(s-u)^b+(r-u)^b]du\right)1_{\{s-r>\varepsilon\}}\\
&=C_{a,b}\left(\int_r^{r+\varepsilon}u^a(s-u)^bdu +(r+\varepsilon)^{1+a+b}-r^{1+a+b} \right)1_{\{s-r>\varepsilon\}}\\
&\leq C_{a,b}\varepsilon^\beta \left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon)^{1+a+b-\beta}\right) 1_{\{s-r>\varepsilon\}}
\end{split}
\end{equation}
for all $1+b<\beta<1$. Moreover, by Lemma~\ref{lem3.4} we have
\begin{align*}
|E[B^{a,b}_s(B^{a,b}_{r+\varepsilon}-B^{a,b}_r)]| 1_{\{s-r<\varepsilon\}}&\leq C_{a,b}s^a\varepsilon^{1+b}1_{\{0<s-r<\varepsilon\}}\\
&\leq C_{a,b}s^a \varepsilon^{\beta}(s-r)^{1+b-\beta}1_{\{0<s-r<\varepsilon\}}
\end{align*}
for all $1+b<\beta<1$. It follows from Lemma~\ref{lem3.4} that
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} &|A_1(s,r,\varepsilon_1,2)|
\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{s}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(\varepsilon_2^{1+b}| E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|+ \varepsilon_1^{1+b}|E[B^{a,b}_{s}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)] |\right)\\
&\leq C_{a,b}s^a
\left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon_1)^{1+a+b-\beta}\right) \varepsilon_1^{\beta-1-b}
1_{\{s-r>\varepsilon_1\}}\\
&\quad+C_{a,b}s^a
\left((r^a\vee s^a)(s-r)^{1+b-\beta} +(r+\varepsilon_2)^{1+a+b-\beta}\right) \varepsilon_2^{\beta-1-b}
1_{\{s-r>\varepsilon_2\}}\\
&\quad+C_{a,b}s^{2a}(s-r)^{1+b-\beta}\varepsilon^{\beta}_1
1_{\{0<s-r<\varepsilon_1\}} +C_{a,b}s^{2a}(s-r)^{1+b-\beta}\varepsilon^{\beta}_2
1_{\{0<s-r<\varepsilon_2\}}\\
&\longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
for all $s>r>0$. Similarly, for all $s>r>0$ and $1+b<\beta<1$ we have also
\begin{align*}
\frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} &|A_3(s,r,\varepsilon_1,2)|\leq \frac1{\varepsilon_1^{2+2b}\varepsilon_2^{1+b}} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]|\\
&\qquad \cdot\left(
\varepsilon_2^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_1}-B^{a,b}_r)]|
+\varepsilon_1^{1+b} |E[B^{a,b}_{r}(B^{a,b}_{r+\varepsilon_2}-B^{a,b}_r)]|\right)\\
&\leq C_{a,b}(s-r)^{b}s^{a-\gamma+1}r^a\varepsilon_1^{\gamma-1-b} \longrightarrow 0\quad (\varepsilon_1,\varepsilon_2\to 0)
\end{align*}
by Lemma~\ref{lem3.4} and using the estimate
\begin{align*} |E[B^{a,b}_{r}(B^{a,b}_{s+\varepsilon_1}-B^{a,b}_s)]&=
\frac1{2{\mathbb B}(1+a,1+b)} \int_0^ru^a[(s-u)^b -({s+\varepsilon_1}-u)^b]du \\
&=\frac1{2{\mathbb B}(1+a,1+b)} \int_0^ru^a\frac{({s+\varepsilon_1}-u)^{-b}-(s-u)^{-b}}
{({s+\varepsilon_1}-u)^{-b}(s-u)^{-b}}du\\
&\leq \frac1{2{\mathbb B}(1+a,1+b)}\varepsilon_1^\gamma
\int_0^r\frac{u^a}
{({s+\varepsilon_1}-u)^{\gamma}(s-u)^{-b}}du\\
&\leq \frac1{2{\mathbb B}(1+a,1+b)}\frac{\varepsilon_1^\gamma}{(s-r)^{-b}}
\int_0^s\frac{u^a}{(s-u)^{\gamma}}du\\ &=C_{a,b}(s-r)^{b}s^{a-\gamma+1}\varepsilon_1^\gamma
\end{align*}
for all $s>r>0,\;(-b)\vee (1+b)<\gamma<1$. Consequently, Lebesgue's dominated convergence theorem implies that the convergence~\eqref{step3-eq1} holds again.
Thus, we have established the convergence~\eqref{sec4-Con-eq1} for $i=1,j=2$ and the theorem follows.
\end{proof}
\begin{corollary}\label{cor5.2}
Let $f,f_1,f_2,\ldots\in {\mathscr H}$. If $f_n\to f$ in ${\mathscr
H}$, as $n$ tends to infinity, then we have
$$
[f_n(B^{a,b}),B^{a,b}]^{(a,b)}_t\longrightarrow [f(B^{a,b}),B^{a,b}]^{(a,b)}_t
$$
in $L^2$ as $n\to \infty$.
\end{corollary}
\begin{proof}
The corollary follows from
$$
E\left|[f_n(B^{a,b}),B^{a,b}]^{(a,b)}_t -[f(B^{a,b}),B^{a,b}]^{(a,b)}_t\right|^2\leq
C_{a,b}\|f_n-f\|_{\mathscr H}^2\to 0,
$$
as $n$ tends to infinity.
\end{proof}
\section{The generalized Bouleau-Yor identity with $b<0$}\label{sec7}
In this section, we study the Bouleau-Yor identity. It is known that the quadratic covariation $[f(B),B]$ of Brownian motion $B$ can be characterized
as
$$
[f(B),B]_t=-\int_{\mathbb R}f(x){\mathscr L}^{B}(dx,t),
$$
where $f$ is locally square integrable, ${\mathscr L}^{B}(x,t)$
is the local time of Brownian motion and the quadratic covariation $\bigl[f(B),B\bigr]$ (see Russo-Vallois~\cite{Russo2}) is defined by
$$
\bigl[f(B),B\bigr]:=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon} \int_0^{t}\left\{ f(B_{s+\varepsilon})
-f(B_s)\right\}(B_{s+\varepsilon}-B_s)ds,
$$
provided the limit exists uniformly in probability. This is called the Bouleau-Yor identity. More works for this can be found in
Bouleau-Yor~\cite{Bouleau}, Eisenbaum~\cite{Eisen1}, F\"ollmer {\it
et al}~\cite{Follmer}, Feng--Zhao~\cite{Feng,Feng3},
Peskir~\cite{Peskir1}, Rogers--Walsh~\cite{Rogers2},
Yan et al~\cite{Yan4,Yan1,Yan2} and the references therein. However, the Bouleau-Yor identity is not true for weighted-fBm with $b<0$ because
$$
[B^{a,b},B^{a,b}]_t=+\infty
$$
for all $t>0$ and $-1<b<0$. In this section, we shall obtain a generalized Bouleau-Yor identity based on the generalized quadratic covariation defined in Section~\ref{sec6}
Recall that for any closed interval $I\subset {\mathbb R}_{+}$ and for any $x\in {\mathbb R}$, the local time $L(x,I)$ of $B^{a,b}$ is defined as
the density of the occupation measure $\mu_I$ defined by
$$
\mu_I(A)=\int_I1_A(B^{a,b}_s)ds
$$
It can be shown (see Geman and Horowitz~\cite{Geman}, Theorem 6.4) that the following occupation density formula holds:
$$
\int_Ig(B^{a,b}_s,s)ds=\int_{\mathbb R}dx\int_Ig(x,s) L(x,ds)
$$
for every Borel function $g(x,t)\geq 0$ on $I\times {\mathbb R}$. Thus, Lemma~\ref{lem3.0} in Section~\ref{sec3} and Theorem 21.9 in Geman-Horowitz~\cite{Geman} together imply that the following result holds.
\begin{corollary}
Let $a>-1,|b|<1,|b|<1+a$ and let $L(x,t):=L(x, [0,t])$ be the local time of $B^{a,b}$ at $x$. Then $L\in L^2(\lambda\times P)$ for all $t\geq 0$ and $(x,t)\mapsto L(x,t)$ is jointly continuous if and only if $a+b<3$, where $\lambda$ denotes Lebesgue measure. Moreover, the occupation formula
\begin{equation}\label{sec2-1-eq1}
\int_0^t\psi(B^{a,b}_s,s)ds=\int_{\mathbb R}dx\int_0^t\psi(x,s) L(x,ds)
\end{equation}
holds for every continuous and bounded function $\psi(x,t):{\mathbb
R}\times {\mathbb R}_{+}\rightarrow {\mathbb R}$ and any $t\geqslant
0$.
\end{corollary}
Define the weighted local time ${\mathscr L}^{a,b}$ by
\begin{align*}
{\mathscr L}^{a,b}(x,t)&=(1+a+b)\int_0^ts^{a+b}d_sL(s,x)\\
&\equiv (1+a+b)\int_0^t\delta(B^{a,b}_s-x)s^{a+b}ds
\end{align*}
for $t\geq 0$ and $x\in {\mathbb R}$, where $\delta$ is the Dirac delta function. In this section, we define the integral
\begin{equation}\label{sec7-eq7.1}
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t),
\end{equation}
for $b<0$, where $f$ is a Borel function. We shall use the generalized quadratic covariation to study it and obtain the following generalized Bouleau-Yor identity:
\begin{equation}\label{Bouleau-Yor}
[f(B^{a,b}),B^{a,b}]_t^{(a,b)}=-\kappa_{a,b}\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{equation}
for all $f\in {\mathscr H}$, $-1<b<0$, $a>-1$, $-1<a+b<3$ and $t\geq 0$. We first give an extension of It\^{o} formula stated as follows.
\begin{theorem}\label{th7.1}
Let $a>-1$, $-1<b<0$, $|b|<1+a$ and let $f\in {\mathscr H}$ be a left continuous function with right limits. If $F$ is an absolutely continuous function with $F'=f$ and
\begin{equation}\label{sec7-Ito-con1}
\max\left\{|F(x)|,|f(x)|\right\}\leq C e^{\beta x^2},
\end{equation}
where $C$ and $\beta$ are some positive constants with
$\beta<\frac14T^{-(1+a+b)}$, then the It\^o type formula
\begin{equation}\label{sec7-eq7.3}
F(B^{a,b}_t)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t
\end{equation}
holds for all $t\in [0,T]$.
\end{theorem}
Clearly, this is an analogue of F\"ollmer-Protter-Shiryayev's
formula F\"ollmer (see, for example, F\"ollmer {\it et al}~\cite{Follmer}). It is an improvement in terms of the
hypothesis on $f$ and it is also quite interesting itself.
Based on the localization argument and smooth approximation one can prove Theorem~\ref{th7.1}. The localization is that one can localize the domain ${\rm Dom}(\delta^{a,b})$ of the operator $\delta^{a,b}$ (see Nualart~\cite{Nualart1}). Suppose that
$\{(\Omega_n, u^{(n)}), n\geq 1\}\subset {\mathscr F}\times {\rm
Dom}(\delta^{a,b})$ is a localizing sequence for $u$, i.e., the
sequence $\{(\Omega_n, u^{(n)}), n\geq 1\}$ satisfies
\begin{itemize}
\item [(i)] $\Omega_n\uparrow \Omega$, a.s.;
\item [(ii)] $u=u^{(n)}$ a.s. on $\Omega_n$.
\end{itemize}
If $\delta^{a,b}(u^{(n)})= \delta^{a,b}(u^{(m)})$ a.s. on $\Omega_n$ for
all $m\geq n$, then, the divergence $\delta^{a,b}$ is the random
variable determined by the conditions
$$
\delta^{a,b}(u)|_{\Omega_n}=\delta^{a,b} (u^{(n)})|_{\Omega_n}\qquad
{\rm { for\;\; all\;\;}} n\geq 1,
$$
which may depend on the localizing sequence.
\begin{lemma}[Nualart~\cite{Nualart1}]\label{complete2.1} Let
$\{v^{(n)}\}$ be a sequence such that $v^{(n)}\to v$ in $L^2$, as $n\to \infty$ and let
$$
\delta^{a,b}(v^{(n)})=\int_0^Tv_s^{(n)}dB^{a,b}_s,\qquad n\geq 1
$$
exist in $L^2$. If $\delta^{a,b}(v^{(n)})\to G$ in $L^2$, then
$\delta^{a,b}(v)=\int_0^Tv_sdB^{a,b}_s$ exists in $L^2$ and equals to $G$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{th7.1}]
If $F\in C^2({\mathbb R})$, then this is It\^o's formula since
$$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=(1+a+b) \kappa_{a,b} \int_0^tf'(B^{a,b}_s)s^{a+b}ds.
$$
by~\eqref{sec6-eq6.2}. The assumption $F\in C^2({\mathbb R})$ is not correct. By a localization argument we may
assume that the function $f$ is uniformly bounded. In fact, for any
$k\geq 0$ we may consider the set
$$
\Omega_k=\left\{\sup_{0\leq t\leq T}|B^{a,b}_t|<k\right\}
$$
and let $f^{[k]}$ be a measurable function such that $f^{[k]}=f$ on
$[-k,k]$ and such that $f^{[k]}$ vanishes outside. Then $f^{[k]}\in
{\mathscr H}$ and uniformly bounded. Set
$\frac{d}{dx}F^{[k]}=f^{[k]}$ and $F^{[k]}=F$ on $[-k,k]$. If the
theorem is true for all uniformly bounded functions on ${\mathscr
H}$, then we get the desired formula
$$
F^{[k]}(B^{a,b}_t)=F^{[k]}(0)+\int_0^t
f^{[k]}(B^{a,b}_s)dB^{a,b}_s+\frac12(\kappa_{a,b})^{-1}
\bigl[f^{[k]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
on the set $\Omega_k$. Letting $k$ tend to infinity we deduce the
It\^o formula~\eqref{sec7-eq7.3} for all $f\in {\mathscr H}$ being
left continuous and locally bounded.
Let now $F'=f\in {\mathscr H}$ be uniformly bounded such that the conditions in the theorem hold. Define the sequence of smooth functions
$$
F_n(x):=\int_{\mathbb R}F(x-{y})\zeta_n(y)dy,\quad x\in {\mathbb R},
$$
where the mollifiers $\zeta_n,n\geq 1$ are given by~\eqref{sec7-eq7.4}. Then $F_n\in C^\infty({\mathbb R})$ and $F_n$ satisfies the condition~\eqref{sec2-Ito-con1}, and the It\^{o} formula
\begin{equation}\label{sec7-eq7.5}
F_n(B^{a,b}_t)=F_n(0)+\int_0^tf_n(B^{a,b}_s)dB^{a,b}_s+
\frac12(1+a+b)\int_0^tf_n'(B^{a,b}_s)s^{a+b}ds
\end{equation}
holds for all $n\geq 1$, where $f_n=F_n'$. We also have
$$
F_n \longrightarrow F,\quad f_n \longrightarrow f,
$$
uniformly in $\mathbb R$, as $n$ tends to infinity, and moreover, $\{f_n\}\subset {\mathscr H}$, $f_n\to f$ in ${\mathscr H}$, as $n$ tends to infinity. It follows that
\begin{align*}
(1+a+b)\int_0^tf_n'(B^{a,b}_s)s^{a+b}ds &=(\kappa_{a,b})^{-1}\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&\longrightarrow (\kappa_{a,b})^{-1}\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{align*}
in $L^2(\Omega)$ by Corollary~\ref{cor5.2}, as $n$ tends to infinity. It follows that
\begin{align*}
\int_0^tf_n(B^{a,b}_s)dB^{a,b}_s&=F_n(B^{a,b}_t)-F_n(0)-
\frac12(\kappa_{a,b})^{-1}\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&\longrightarrow F(B^{a,b}_t)-F(0)-\frac12(\kappa_{a,b})^{-1} \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{align*}
in $L^2(\Omega)$, as $n$ tends to infinity. This completes the proof since
the integral is closed in $L^2(\Omega)$.
\end{proof}
Now, let us study the integral~\eqref{sec7-eq7.1} for $-1<b<0$.
\begin{lemma}
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $f_\triangle=\sum_jf_j1_{(a_{j-1},a_j]}\in {\mathscr H}$, we
define
$$
\int_{\mathbb R}f_\triangle(x){\mathscr
L}^{a,b}(dx,t):=\sum_jf_j\left[{\mathscr L}^{a,b}(a_j,t)-{\mathscr
L}^{a,b}(a_{j-1},t)\right].
$$
Then the integral is well-defined and
\begin{equation}\label{sec6-eq6.4}
\kappa_{a,b}\int_{\mathbb R}f_{\Delta}(x)\mathscr{L}^{a,b}(dx,t)=
-\bigl[f_\triangle(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
\end{equation}
almost surely, for all $t\in [0,T]$.
\end{lemma}
\begin{proof}
For the function $f_\triangle(x)=1_{(a,b]}(x)$ we define the sequence of smooth functions $f_n,\;n=1,2,\ldots$ by
\begin{align}
f_n(x)&=\int_{\mathbb
R}f_\triangle(x-y)\zeta_n(y)dy=\int_a^b\zeta_n(x-u)du
\end{align}
for all $x\in \mathbb R$, where $\zeta_n,n\geq 1$ are the so-called
mollifiers given in~\eqref{sec7-eq7.4}. Then $\{f_n\}\subset
C^{\infty}({\mathbb R})\cap {\mathscr H}$ and $f_n$ converges to $f_\triangle$ in ${\mathscr H}$, as $n$ tends to infinity. It follows from the occupation formula that
\begin{align*}
\bigl[f_n(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t& =\kappa_{a,b}\int_0^tf'_n(B^{a,b}_s)ds^{1+a+b}\\
&=\kappa_{a,b}\int_{\mathbb R}f_n'(x){\mathscr L}^H(x,t)dx=\int_{\mathbb
R}\left(\int_a^b\zeta_n'(x-u)du\right){\mathscr L}^{a,b}(x,t)dx\\
&=-\kappa_{a,b}\int_{\mathbb R}{\mathscr
L}^{a,b}(x,t)\left(\zeta_n(x-b)-\zeta_n(x-a)\right)dx\\
&=\kappa_{a,b}\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)\zeta_n(x-a)dx
-\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)\zeta_n(x-b)dx\\
&\longrightarrow \kappa_{a,b}\left({\mathscr L}^{a,b}(a,t)-{\mathscr L}^{a,b}(b,t)\right)
\end{align*}
almost surely, as $n\to \infty$, by the continuity of $x\mapsto
{\mathscr L}^{a,b}(x,t)$. On the other hand, Corollary~\ref{cor5.2}
implies that there exists a subsequence $\{f_{n_k}\}$ such that
$$
\bigl[f_{n_k}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\longrightarrow \bigl[1_{(a,b]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
for all $t\in [0,T]$, almost surely, as $k\to \infty$. Then we have
$$
\bigl[1_{(a,b]}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t =\kappa_{a,b}\left({\mathscr L}^{a,b}(a,t)-{\mathscr L}^{a,b}(b,t)
\right)
$$
for all $t\in [0,T]$, almost surely. Thus, the identity
$$
\kappa_{a,b}\sum_jf_j\bigl[{\mathscr L}^{a,b}(a_j,t)-{\mathscr
L}^{a,b}(a_{j-1},t)\bigr]=
-\bigl[f_\triangle(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t
$$
follows from the linearity property, and the lemma follows.
\end{proof}
As a direct consequence of the above lemma we can show
that
\begin{equation} \lim_{n\to \infty}\int_{\mathbb
R}f_{\triangle,n}(x){\mathscr L}^{a,b}(x,t)dx =\lim_{n\to
\infty}\int_{\mathbb R}g_{\triangle,n}(x){\mathscr L}^{a,b}(x,t)dx
\end{equation}
in $L^2(\Omega)$ if
$$
\lim_{n\to \infty}f_{\triangle,n}(x)=\lim_{n\to
\infty}g_{\triangle,n}(x)=f(x)
$$
in ${\mathscr H}$, where $\{f_{\triangle,n}\},\{g_{\triangle,n}\}\subset
{\mathscr E}$. Thus, by the density of ${\mathscr
E}$ in ${\mathscr H}$ we can define
$$
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t):=\lim_{n\to
\infty}\int_{\mathbb R}f_{\triangle,n}(x){\mathscr L}^{a,b}(dx,t)
$$
for any $f\in {\mathscr H}$, where $\{f_{\triangle,n}\}\subset
{\mathscr E}$ and
$$
\lim_{n\to \infty}f_{\triangle,n}(x)=f(x)
$$
in ${\mathscr H}$. These considerations are enough to prove the following theorem.
\begin{theorem}\label{th7.2}
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $f\in {\mathscr H}$, the integral
$$
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
$$
is well-defined in $L^2(\Omega)$ and the Bouleau-Yor type formula
\begin{equation}\label{sec6-eq6.3}
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=-\kappa_{a,b}\int_{\mathbb R}f(x)\mathscr{L}^{a,b}(dx,t)
\end{equation}
holds, almost surely, for all $t\in [0,T]$.
\end{theorem}
\begin{corollary}[Tanaka formula]
Let $-1<b<0$, $a>-1$ and $-1<a+b<3$. For any $x\in {\mathbb R}$ we have
\begin{align*}
(B^{a,b}_t-x)^{+}=(-x)^{+}+\int_0^t{1}_{\{B^{a,b}_s>x\}} dB^{a,b}_s
+\frac12{\mathscr L}^{a,b}(x,t),\\
(B^{a,b}_t-x)^{-}=(-x)^{-}-\int_0^t{1}_{\{B^{a,b}_s<x\}}
dB^{a,b}_s+\frac12{\mathscr L}^{a,b}(x,t),\\
|B^{a,b}_t-x|=|x|+\int_0^t{\rm sign}(B^{a,b}_s-x)dB^{a,b}_s+{\mathscr L}^{a,b}(x,t).
\end{align*}
\end{corollary}
\begin{proof}
Take $F(y)=(y-x)^{+}$. Then $F$ is absolutely continuous and
$$
F(x)=\int_{-\infty}^y1_{(x,\infty)}(y)dy.
$$
It follows from It\^o's formula~\eqref{sec7-eq7.3} and the
identity~\eqref{sec6-eq6.4} that
\begin{align*}
{\mathscr
L}^{a,b}(x,t)&=(\kappa_{a,b})^{-1} \bigl[1_{(x,+\infty)}(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t\\
&=2(B^{a,b}_t-a)^{+}-2(-x)^{+}-2 \int_0^t{1}_{\{B^{a,b}_s>x\}}dB^{a,b}_s
\end{align*}
for all $t\in [0,T]$, which gives the first identity. In the same
way one can obtain the second identity. By subtracting the last
identity from the previous one, we get the third identity.
\end{proof}
\begin{corollary}
Let $-1<b<0$, $a>-1$, $-1<a+b<3$ and let $f,f_1,f_2,\ldots\in {\mathscr H}$. If $f_n\to f$ in ${\mathscr H}$, as $n$ tends to infinity, we then have
\begin{align*}
\int_{\mathbb R}f_n(x){\mathscr L}^{a,b}(dx,t)\longrightarrow
\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t)
\end{align*}
in $L^2$, as $n$ tends to infinity.
\end{corollary}
According to Theorem~\ref{th7.1}, we get an analogue of the It\^o
formula (Bouleau-Yor type formula).
\begin{corollary}\label{cor7.1}
Let $-1<b<0$, $a>-1$, $-1<a+b<3$ and let $f\in {\mathscr H}$ be a left continuous function with right limits. If $F$ is an absolutely continuous function with $F'=f$ and the condition~\eqref{sec7-Ito-con1} is satisfied, then the following It\^o type formula holds:
\begin{equation}\label{sec7-eq7.11}
F(B^{a,b}_t)=F(0)+\int_0^tf(B^{a,b}_s)dB^{a,b}_s -\frac12\int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t).
\end{equation}
\end{corollary}
Recall that if $F$ is the difference of two convex functions, then
$F$ is an absolutely continuous function with derivative of bounded
variation. Thus, the It\^o-Tanaka formula
\begin{align*}
F(B^{a,b}_t)&=F(0)+\int_0^tF^{'}(B^{a,b}_s)dB^{a,b}_s +\frac12\int_{\mathbb R}{\mathscr L}^{a,b}(x,t)F''(dx)\\
&\equiv F(0)+\int_0^tF^{'}(B^{a,b}_s)dB^{a,b}_s -\frac12\int_{\mathbb R}F'(x){\mathscr L}^{a,b}(dx,t)
\end{align*}
holds for all $-1<b<0$, $a>-1$ and $-1<a+b<3$.
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