diff --git "a/CdE1T4oBgHgl3EQfDwOw/content/tmp_files/load_file.txt" "b/CdE1T4oBgHgl3EQfDwOw/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/CdE1T4oBgHgl3EQfDwOw/content/tmp_files/load_file.txt" @@ -0,0 +1,472 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf,len=471 +page_content='MLMC techniques for discontinuous functions Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles Abstract The Multilevel Monte Carlo (MLMC) approach usually works well when estimating the expected value of a quantity which is a Lipschitz function of inter- mediate quantities, but if it is a discontinuous function it can lead to a much slower decay in the variance of the MLMC correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This article reviews the literature on techniques which can be used to overcome this challenge in a variety of different contexts, and discusses recent developments using either a branching diffusion or adaptive sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 1 Introduction The Multilevel Monte Carlo (MLMC) method is based on the telescoping sum E[ �𝑃𝐿] = E[ �𝑃0] + 𝐿 βˆ‘οΈ β„“=1 E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1] where �𝑃ℓ represents an approximation to an output quantity of interest 𝑃 on level β„“, with the weak error οΏ½οΏ½οΏ½E[ οΏ½π‘ƒβ„“βˆ’π‘ƒ] οΏ½οΏ½οΏ½ and MLMC variance V[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1], both decreasing as the level β„“ increases, but with the corresponding computational cost per sample increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If οΏ½π‘Œβ„“ has expected value E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1], with variance 𝑉ℓ and cost 𝐢ℓ, then for a given target RMS error πœ€, the number of levels 𝐿 and the number of independent samples on each level can be optimised [13, 14] to give an overall cost which is approximately equal to πœ€βˆ’2 οΏ½ 𝐿 βˆ‘οΈ β„“=0 √︁ 𝐢ℓ𝑉ℓ οΏ½2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles University of Oxford Mathematical Institute, Woodstock Rd, Oxford OX2 6GG, UK e-mail: mike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='giles@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='02882v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='NA] 7 Jan 2023 2 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles If 𝐢ℓ𝑉ℓ β†’ 0 as β„“ β†’ ∞, then the cost is dominated by the first term from level 0, and the cost is approximately πœ€βˆ’2𝐢0𝑉0, so proportional to πœ€βˆ’2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝐢ℓ𝑉ℓ β†’ const as β„“ β†’ ∞, then the contributions to the cost are spread almost equally across all levels and the cost is approximately πœ€βˆ’2𝐿2𝐢𝐿𝑉𝐿, proportional to πœ€βˆ’2| log πœ€|2 if E[ οΏ½π‘ƒβ„“βˆ’π‘ƒ] decays exponentially with β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Even worse, if 𝐢ℓ𝑉ℓ β†’ ∞ then the cost is dominated by the contribution from the finest level and so is approximately πœ€βˆ’2𝐢𝐿𝑉𝐿 which is 𝑂(πœ€βˆ’2βˆ’(π›Ύβˆ’π›½)/𝛼) if E[ οΏ½π‘ƒβ„“βˆ’π‘ƒ] ∼ 2βˆ’π›Όβ„“, 𝑉ℓ ∼ 2βˆ’π›½β„“ and 𝐢ℓ ∼ 2𝛾ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In most MLMC applications, 𝑃 is a smooth function of some intermediate solution quantities, such as the solution of an SDE, a PDE with stochastic coefficients or initial/boundary data, or an estimate of an inner conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Under these circumstances we usually have 𝛽 β‰₯ 𝛾 and so the MLMC complexity is 𝑂(πœ€βˆ’2) or 𝑂(πœ€βˆ’2| log πœ€|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This article is concerned with the small but important class of applications where 𝑃 is a discontinuous function of the intermediate quantities, and because of this the MLMC variance 𝑉ℓ can decay much more slowly, leading to the complexity falling into the third category of being 𝑂(πœ€βˆ’2βˆ’(π›Ύβˆ’π›½)/𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The good news is that there has been considerable research within the MLMC community to address this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This article surveys the variety of methods which have been developed in the hope that this can aid researchers meeting similar challenges in future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' To illustrate things, we begin by detailing two specific application challenges which have motivated much of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' We then discuss the many approaches which have been developed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' several of which have borrowed ideas from the literature on computing sensitivities (the β€œgreeks” in mathematical finance literature) of the form πœ• πœ•π›ΌE [ 𝑓 (πœ”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 𝛼)] using the pathwise sensitivity approach [24] (also known as Infinitesimal Perturba- tion Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' IPA for short [30]) or Likelihood Ratio Method [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' or from methods from improving integrand smoothness to improve the rate of convergence for QMC integration [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='1 Challenge 1: nested expectation Suppose 𝑓 is a scalar function and we want to estimate the nested expectation E [ 𝑓 (E[𝑍|𝑋]) ], where the outer expectation is with respect to a random variable 𝑋 and we will assume that the inner conditional expectation E[𝑍|𝑋] has a bounded density near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' A very simple MLMC treatment 1 uses 𝑀ℓ = 2ℓ𝑀0 inner samples on level β„“, so estimators on level 0 and the higher levels are simply 1 Note that if 𝑓 is smooth, or at least Lipschitz, then it is better to use an β€œantithetic” estimator [8, 14, 15, 18], but this does not give a better order of convergence when 𝑓 is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' MLMC techniques for discontinuous functions 3 οΏ½π‘Œ0 = 𝑓 (𝑍 (0,𝑀0)), οΏ½π‘Œβ„“ = 𝑓 (𝑍 (β„“,𝑀ℓ)) βˆ’ 𝑓 (𝑍 (β„“,π‘€β„“βˆ’1)), where 𝑍 (β„“,𝑀ℓ) and 𝑍 (β„“,π‘€β„“βˆ’1) represent independent averages of 𝑀ℓ and π‘€β„“βˆ’1 inde- pendent samples of 𝑍, all conditional on the same value of 𝑋 [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If V[𝑍|𝑋] is finite, and 𝑓 is Lipschitz with constant 𝐿 𝑓 , then E οΏ½οΏ½ 𝑓 (𝑍 (β„“,𝑀ℓ)) βˆ’ 𝑓 (E[𝑍|𝑋]) οΏ½2 | 𝑋 οΏ½ ≀ 𝐿2 𝑓 E οΏ½οΏ½ 𝑍 (β„“,𝑀ℓ) βˆ’ E[𝑍|𝑋] οΏ½2 | 𝑋 οΏ½ = 𝐿2 𝑓 π‘€βˆ’1 β„“ V[𝑍|𝑋], and hence E[οΏ½π‘Œ2 β„“ |𝑋] ≀ 4 𝐿2 𝑓 (π‘€βˆ’1 β„“ + π‘€βˆ’1 β„“βˆ’1)V[𝑍|𝑋] for β„“>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If V[𝑍|𝑋] is uniformly bounded it follows that 𝑉ℓ = 𝑂(π‘€βˆ’1 β„“ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If the cost of each conditional sample of 𝑍 is 𝑂(1) then 𝐢ℓ = 𝑂(𝑀ℓ) and hence the complexity is 𝑂(πœ€βˆ’2| log πœ€|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Unfortunately, the situation is significantly poorer when 𝑓 is the Heaviside step function 𝐻 defined by 𝐻(π‘₯)=0 if π‘₯<0, and 𝐻(π‘₯)=1 if π‘₯β‰₯0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This occurs in many applications because P [E[𝑍|𝑋] > 𝐾] = E [𝐻(E[𝑍|𝑋] βˆ’ 𝐾)] , so it corresponds to the probability of a conditional expectation exceeding some threshold 𝐾, which is a very important quantity in risk calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝐾=0 and 𝐸[𝑍|𝑋] has a bounded density near zero then there is an 𝑂(π‘€βˆ’1/2 β„“ ) probability that |𝐸[𝑍|𝑋] | = 𝑂(π‘€βˆ’1/2 β„“ ), which is the circumstance under which there is an 𝑂(1) probability that οΏ½π‘Œβ„“ = Β±1 due to 𝑍 (β„“,𝑀ℓ) being positive and 𝑍 (β„“,π‘€β„“βˆ’1) being negative, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Hence 𝑉ℓ β‰ˆ 𝑂(π‘€βˆ’1/2 β„“ ) and the complexity is approximately 𝑂(πœ€βˆ’5/2) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This challenge is the primary motivation for Section 7, and also arises in the context of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 Challenge 2: discontinuous payoff function In the case of a scalar SDE d𝑆𝑑 = π‘Ž(𝑆𝑑) d𝑑 + 𝑏(𝑆𝑑) dπ‘Šπ‘‘, (1) with an output quantity of interest 𝑃 ≑ 𝑓 (𝑆𝑇 ), the standard estimator is οΏ½π‘Œβ„“ = �𝑃ℓ βˆ’ οΏ½π‘ƒβ„“βˆ’1 where the same Brownian motion π‘Šπ‘‘ is used to calculate both �𝑃ℓ and οΏ½π‘ƒβ„“βˆ’1, but with different uniform timesteps β„Žβ„“ and β„Žβ„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝑓 is Lipschitz with constant 𝐿 𝑓 , then 𝑉ℓ ≀ E οΏ½ ( �𝑃ℓ βˆ’ οΏ½π‘ƒβ„“βˆ’1)2οΏ½ ≀ 𝐿2 𝑓 E οΏ½ (�𝑆ℓ βˆ’ οΏ½π‘†β„“βˆ’1)2οΏ½ 4 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles where �𝑆ℓ is the level β„“ numerical approximation to 𝑆𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Hence, based on standard strong convergence results [28] we have 𝑉ℓ = 𝑂(β„Žβ„“) for an Euler-Maruyama dis- cretisation of the SDE, and 𝑉ℓ = 𝑂(β„Ž2 β„“) for the first order Milstein discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The cost 𝐢ℓ is 𝑂(β„Žβˆ’1 β„“ ) in both cases, giving MLMC complexities of 𝑂(πœ€βˆ’2| log πœ€|2) and 𝑂(πœ€βˆ’2), respectively, In mathematical finance, a digital call option payoff is 0 or 1, depending on whether 𝑆𝑇 is below or above the strike 𝐾, so the payoff function can be written as 𝑓 (𝑆𝑇 ) = 𝐻(𝑆𝑇 βˆ’πΎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The MLMC problem is that a small difference between the coarse and fine paths can give a payoff difference of Β±1 if the two paths straddle the strike, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' are on different sides of the strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' When using the Euler-Maruyama approximation of the SDE, οΏ½π‘†β„“βˆ’οΏ½π‘†β„“βˆ’1 = 𝑂(β„Ž1/2 β„“ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Speaking loosely (see [4, 21] for the rigorous analysis) an 𝑂(β„Ž1/2 β„“ ) fraction of fine/coarse pairs straddle the strike, so 𝑉ℓ = 𝑂(β„Ž1/2 β„“ ) and hence the complexity is 𝑂(πœ€βˆ’5/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Similarly, using the Milstein approximation gives οΏ½π‘†β„“βˆ’οΏ½π‘†β„“βˆ’1 = 𝑂(β„Žβ„“) so 𝑉ℓ = 𝑂(β„Žβ„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This is clearly better, and gives a complexity which is 𝑂(πœ€βˆ’2| log πœ€|2), but there is still the problem that most MLMC samples π‘Œβ„“ are zero on the finer levels, so the kurtosis is 𝑂(β„Žβˆ’1 β„“ ) which causes problems in practice in estimating 𝑉ℓ accurately to determine the number of samples 𝑁ℓ to use on level β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In addition, there is the difficulty that the Milstein discretisation of multi-dimensional SDEs often requires the simulation of LΓ©vy areas, though this problem can be addressed through the use of an antithetic estimator [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This challenge is the primary motivation for Sections 2, 4, 5 and 6, also also arises in Sections 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 2 Explicit smoothing The pathwise sensitivity analysis (or IPA) approach to compute the parameter sen- sitivities known as Greeks in mathematical finance [24] requires that the payoff function 𝑓 is continuous and piecewise smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This is clearly a problem with digi- tal options, and one standard approach is to smooth the payoff function by replacing the Heaviside step function 𝐻 with a smoothed approximation 𝐻𝛿(π‘₯) ≑ 𝑔(π‘₯/𝛿), with 𝑔(π‘₯) β†’ 0 as π‘₯ β†’ βˆ’βˆž and 𝑔(π‘₯) β†’ 1 as π‘₯ β†’ +∞, so the discontinuity is smoothed over a width of 𝑂(𝛿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For financial reasons, the preference is often to use a one-sided smoothing, such as the piecewise linear approximation shown in yellow in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This one-sided approximation introduces a weak error, or bias, which is 𝑂(𝛿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If it is used for MLMC, then 𝐻′ 𝛿(𝑆𝑇 ) = π›Ώβˆ’1 for the 𝑂(𝛿) fraction of the paths which end up in the ramp region, and therefore 𝑉ℓ = 𝑂(𝛿 Γ— (π›Ώβˆ’1)2) = 𝑂(π›Ώβˆ’1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Hence the optimal choice of 𝛿 involves a tradeoff between bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The bias can be reduced by making the smoothing anti-symmetric about π‘₯ = 0 so that 𝐻𝛿(π‘₯) βˆ’ 𝐻(π‘₯) = βˆ’(𝐻𝛿(βˆ’π‘₯) βˆ’ 𝐻(βˆ’π‘₯)), for example by choosing 𝑔(π‘₯) ≑ Ξ¦(π‘₯) MLMC techniques for discontinuous functions 5 as illustrated in orange in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝑆𝑇 has the smooth probability density 𝜌(𝑆) then the weak error is ∫ ∞ βˆ’βˆž (𝐻𝛿(π‘†βˆ’πΎ) βˆ’ 𝐻(π‘†βˆ’πΎ)) 𝜌(𝑆) d𝑆 = 𝛿 ∫ ∞ βˆ’βˆž (𝑔(π‘₯) βˆ’ 𝐻(π‘₯)) 𝜌(𝐾+π‘₯𝛿) dπ‘₯ and a Taylor series expansion of 𝜌(𝐾+π‘₯𝛿) results in the asymptotic error expansion π‘Ž1𝜌(𝐾) 𝛿 + π‘Ž2πœŒβ€²(𝐾) 𝛿2 + π‘Ž3πœŒβ€²β€²(𝐾) 𝛿3 + π‘Ž4πœŒβ€²β€²β€²(𝐾) 𝛿4 + 𝑂(𝛿5) where π‘Žπ‘˜ = ∫ ∞ βˆ’βˆž π‘₯π‘˜βˆ’1 (𝑔(π‘₯) βˆ’ 𝐻(π‘₯)) dπ‘₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝑔(π‘₯) βˆ’ 𝐻(π‘₯) = βˆ’ (𝑔(βˆ’π‘₯) βˆ’ 𝐻(βˆ’π‘₯)) then π‘Ž1 = π‘Ž3 = 0, and π‘Ž2 = 2 ∫ ∞ 0 π‘₯(𝑔(π‘₯) βˆ’ 1) dπ‘₯, π‘Ž4 = 2 ∫ ∞ 0 π‘₯3(𝑔(π‘₯) βˆ’ 1) dπ‘₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝑔(π‘₯) is monotonic, then π‘Ž2 β‰  0, but by considering non-monotonic functions such as 𝑔(π‘₯) = (4/3) Ξ¦(π‘₯) βˆ’ (1/3) Ξ¦(2π‘₯) it is possible to set π‘Ž2 = 0 making the weak error 𝑂(𝛿4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Hence, to achieve 𝑂(πœ€) accuracy overall we need 𝛿=𝑂(πœ€1/4), and then on the coarsest levels 𝑉ℓ = 𝑂(π›Ώβˆ’1) = 𝑂(πœ€βˆ’1/4) so the overall complexity is approximately 𝑂(πœ€βˆ’2βˆ’1/4) in the best cases where the overall cost is dominated by the cost on the coarsest levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles, Nagapetyan & Ritter [22] used explicit smoothing for estimating cumulative distribution functions (CDFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For a scalar random variable 𝑋, to estimate 𝐢(π‘₯) = P(𝑋 < π‘₯) = E[𝐻(π‘₯βˆ’π‘‹)], the approach they adopted was to use MLMC to estimate 𝐢(π‘₯ 𝑗) for a set of spline points π‘₯ 𝑗 and then interpolate these values with a cubic spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Overall, their method balanced three weak errors, the SDE discretisation error on the finest level, the smoothing error due to 𝐻𝛿, and the cubic spline interpolation error, in addition to the MLMC sampling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 ST 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 f(S T) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 1 Two explicitly smoothed versions of the Heaviside step function for a digital call option 6 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles 3 Integration/differentiation and Malliavin calculus Krumscheid & Nobile [29] used a slightly different approach for estimating CDFs, particularly in the context of risk estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Starting from the identity d dπ‘₯ E [ max(0, π‘₯βˆ’π‘†π‘‡ ) ] = E[ 𝐻(π‘₯βˆ’π‘†π‘‡ ) ] they used MLMC to estimate E[ max(0, π‘₯ π‘—βˆ’π‘†π‘‡ ) ] for a set of spline points π‘₯ 𝑗, interpolated these with a cubic spline, and and then differentiated the spline to obtain an approximation to the CDF 𝐢(π‘₯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This avoids the extra weak error due to smoothing the Heaviside function, but differentiating the cubic spline amplifies the noise in the spline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' On a similar note, Altmayer & Neuenkirch [2] used Malliavin calculus integration by parts to treat discontinuous payoffs based on solutions of the Heston stochastic volatility SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' They observed that asymptotically this improves the MLMC vari- ance on the finer levels, but it increases the variance on coarse levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' To address this, they split the payoff into a smooth part which they treated with the standard MLMC approach, and a compact-support discontinuous part for which they used the Malliavin MLMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Malliavin calculus was originally developed for computing sensitivities, so this is another example of the literature on sensitivity calculations being exploited to develop improved MLMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 4 Conditional expectation When using the first order Milstein discretisation for an SDE, one way to improve the MLMC variance for digital options is to switch to the Euler-Maruyama approxi- mation for the final timestep, and then take the conditional expectation with respect to the final fine path Brownian increment Ξ”π‘Š [12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For the fine path approximation of the scalar SDE (1) with 𝑁 timesteps of size β„Žβ„“, the path value 𝑆𝑇 at the final time 𝑇 is given by �𝑆 𝑓 𝑇 = �𝑆 𝑓 𝑇 βˆ’β„Žβ„“ + π‘Ž(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) β„Žβ„“ + 𝑏(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) Ξ”π‘Šπ‘ , and therefore the conditional expected value for the digital call option is �𝑃 𝑓 β„“ = E οΏ½ 𝐻(�𝑆 𝑓 𝑇 βˆ’πΎ) | �𝑆 𝑓 𝑇 βˆ’β„Žβ„“ οΏ½ = Ξ¦ οΏ½οΏ½ οΏ½ �𝑆 𝑓 𝑇 βˆ’β„Žβ„“ + π‘Ž(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) β„Žβ„“ βˆ’ 𝐾 𝑏(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) βˆšβ„Žβ„“ οΏ½οΏ½ οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' for the coarse path with coarse timestep β„Žβ„“βˆ’1 = 2 β„Žβ„“,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' the Brownian incre- ment for the final coarse timestep is the sum of the last two Brownian increments for the fine path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Ξ”π‘Šπ‘ βˆ’1+Ξ”π‘Šπ‘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' and therefore MLMC techniques for discontinuous functions 7 �𝑆𝑐 𝑇 = �𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1 + π‘Ž(�𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1) β„Žβ„“βˆ’1 + 𝑏(�𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1) (Ξ”π‘Šπ‘ βˆ’1+Ξ”π‘Šπ‘ ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' from which we obtain �𝑃𝑐 β„“βˆ’1 = E οΏ½ 𝐻(�𝑆𝑐 𝑇 βˆ’πΎ) | �𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Ξ”π‘Šπ‘ βˆ’1 οΏ½ = Ξ¦ οΏ½ �𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1 + π‘Ž(�𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1) β„Žβ„“βˆ’1 + 𝑏(�𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1) Ξ”π‘Šπ‘ βˆ’1 βˆ’ 𝐾 𝑏(�𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1) βˆšβ„Žβ„“ οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' With οΏ½π‘Œβ„“ ≑ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1, numerical analysis [17] proves that 𝑉ℓ β‰ˆ 𝑂(β„Ž3/2 β„“ ) so the MLMC complexity is 𝑂(πœ€βˆ’2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Heuristically, this is because there is an 𝑂(β„Ž1/2 β„“ ) probability of paths being within 𝑂(β„Ž1/2 β„“ ) of the strike 𝐾, and for these �𝑆 𝑓 𝑇 βˆ’β„Žβ„“βˆ’1 βˆ’ �𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1 = 𝑂(β„Žβ„“), πœ• �𝑃 πœ•οΏ½π‘† = 𝑂(β„Žβˆ’1/2 β„“ ) =β‡’ �𝑃ℓ βˆ’ οΏ½π‘ƒβ„“βˆ’1 = 𝑂(β„Ž1/2 β„“ ), so 𝑉ℓ β‰ˆ 𝑂(β„Ž1/2 β„“ Γ— (β„Ž1/2 β„“ )2) = 𝑂(β„Ž3/2 β„“ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In addition, the kurtosis is improved to 𝑂(β„Žβˆ’1/2 β„“ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Unfortunately, the conditional expectation approach does not help when the Euler- Maruyama discretisation is used for the entire path since �𝑆 𝑓 𝑇 βˆ’β„Žβ„“βˆ’1βˆ’οΏ½π‘†π‘ 𝑇 βˆ’β„Žβ„“βˆ’1 = 𝑂(β„Ž1/2 β„“ ) and so �𝑃ℓ βˆ’ οΏ½π‘ƒβ„“βˆ’1 = 𝑂(1) The use of this kind of conditional expectation is a standard technique for smooth- ing the payoff to enable IPA/pathwise sensitivity calculations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Another example is a down-and-out barrier option, where the option is knocked out if the path drops below a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In this case the payoff can be smoothed by computing the probability of this happening, conditional on the computed path approximations at discrete timesteps [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Again, this works well for MLMC when using the first or- der Milstein discretisation [12, 17], but it does not help with the Euler-Maruyama discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' A different kind of conditional expectation smoothing was introduced by Achtsis, Cools & Nuyens [1] and Bayer, Siebenmorgen & Tempone [6] to improve the convergence of QMC computations, and then used by Bayer, Ben Hammouda & Tempone [5] to improve the MLMC variance for digital options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In its simplest form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' they split the random inputs for the numerical simulation into a scalar 𝑍 and the remainder π‘π‘Ÿ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' and express the desired MLMC level β„“ expectation as E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1] = E οΏ½ E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 | π‘π‘Ÿ] οΏ½ and observe that in many financial applications it is possible to perform this split in a way such that the conditional expectations E[ �𝑃ℓ | π‘π‘Ÿ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' E[ οΏ½π‘ƒβ„“βˆ’1 | π‘π‘Ÿ] are smooth functions of π‘π‘Ÿ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' and can be evaluated analytically or very accurately by 1D numerical quadrature when there is just a single discontinuity with respect to changes in 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 8 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles For a scalar SDE, 𝑍 could be the terminal value of the driving Brownian motion, in which case π‘π‘Ÿ would represent the other Normal random variables required for a Brownian Bridge construction of the Brownian increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 5 Change of measure Another approach to treating digital options using the Milstein discretisation is to use a change of measure [9, 14], which has connections to the Likelihood Ratio Method (LRM) that is used for sensitivity analysis [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For both the fine and coarse paths, we have conditional Gaussian distributions for �𝑆𝑇 , with slightly different means and variances, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' We can therefore perform a change of measure to the same Gaussian distribution with mean πœ‡ and variance 𝜎2, also illustrated in Figure 2, and then pick the same sample �𝑆𝑇 for both paths from this common Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The resulting MLMC estimator is οΏ½π‘Œβ„“ = 𝑓 (�𝑆𝑇 ) (𝑅ℓ βˆ’ π‘…β„“βˆ’1) where 𝑅ℓ, π‘…β„“βˆ’1 are the respective Radon-Nikodym derivatives for the fine and coarse paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For the scalar SDE (1), 𝑅ℓ is 𝑅ℓ = 𝜎 𝑏(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“)βˆšβ„Žβ„“ exp οΏ½ βˆ’ (�𝑆𝑇 βˆ’ �𝑆 𝑓 𝑇 βˆ’β„Žβ„“ βˆ’ π‘Ž(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) β„Žβ„“)2 / (2 𝑏2(�𝑆 𝑓 𝑇 βˆ’β„Žβ„“) β„Žβ„“) οΏ½ exp οΏ½ βˆ’ (�𝑆𝑇 βˆ’πœ‡)2 / (2𝜎2) οΏ½ and π‘…β„“βˆ’1 is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' It can be shown that the difference 𝑅ℓ βˆ’ π‘…β„“βˆ’1 is approximately 𝑂(β„Ž1/2 β„“ ), which implies that 𝑉ℓ β‰ˆ 𝑂(β„Žβ„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' To improve the variance we note that the conditional expected value of Radon-Nikodym derivatives is always 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 ST 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 p(ST) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 2 Coarse and fine path conditional Gaussian distributions, plus third common distribution MLMC techniques for discontinuous functions 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 t 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 2 S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 3 Illustration of coarse and fine jump-diffusion paths with jumps before and after 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' E[𝑅ℓ | �𝑆 𝑓 𝑇 βˆ’β„Žβ„“] = E[π‘…β„“βˆ’1 | �𝑆𝑐 𝑇 βˆ’β„Žβ„“βˆ’1, Ξ”π‘Šπ‘ βˆ’1] = 1, and therefore we can change the definition of οΏ½π‘Œβ„“ to οΏ½π‘Œβ„“ = οΏ½ 𝑓 (�𝑆𝑇 ) βˆ’ 𝑓 (πœ‡) οΏ½ (𝑅ℓ βˆ’ π‘…β„“βˆ’1) without changing its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This estimator is now non-zero only when �𝑆𝑇 and πœ‡ are on opposite sides of the strike 𝐾, which occurs for an 𝑂(β„Ž1/2 β„“ ) fraction of coarse/fine paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Hence the new MLMC variance 𝑉ℓ is approximately 𝑂(β„Ž3/2 β„“ ), as with the use of the analytic conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The benefit of this approach is that it works well in multiple dimensions when it is often not possible to evaluate the analytic conditional expectation [9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' However, again it does not help with the full path Euler-Maruyama discretisation because that gives 𝑅ℓ βˆ’ π‘…β„“βˆ’1 = 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' An earlier use of a change of measure in an MLMC computation was by Xia [33, 34] for a Merton-style jump-diffusion SDE with a path-dependent jump rate πœ†(𝑆, 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The challenge in this application, as illustrated in Figure 3, is that the coarse and fine paths will jump at different times, and one might jump just before the final time 𝑇, and the other just after, leading to a large jump in the computed value of 𝑓 (𝑆𝑇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The path-dependent jump rate was treated by using the thinning technique of Glasserman & Merener [25], over-sampling possible jump times using a uniform rate πœ†π‘ π‘’π‘ > πœ†(𝑆, 𝑑) and then using an acceptance/rejection step to select the real jump times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Xia modified this with a change of measure to ensure the same acceptance/rejection decision for both the fine and coarse paths so that they both jump at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This leads to an estimator of the form οΏ½π‘Œβ„“ = �𝑃ℓ 𝑅ℓ βˆ’ οΏ½π‘ƒβ„“βˆ’1 π‘…β„“βˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 10 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles When combined with a first order Milstein discretisation of the SDE between the jump times, this gives 𝑉ℓ = 𝑂(β„Ž2 β„“) for Lipschitz payoff functions such as a standard put or call option [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 6 Splitting Returning to the challenge of digital options arising from the solution of an SDE, a third approach is to use path-splitting to generate an unbiased estimate of the conditional expectation introduced in Section 4 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This is a variant of the general splitting technique [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' As illustrated in Figure 4, it involves performing a standard fine path simulation up until one timestep before the final time 𝑇, and then performing multiple independent simulations of the final timestep, averaging the payoff for each of these to get an approximation of the conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The same is done for the coarse path except that each of the splits uses the same Ξ”π‘Šπ‘ βˆ’1 that was used for the second to last fine path timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Since the computational cost of the path up to the splitting time is 𝑂(β„Žβˆ’1 β„“ ), it means that up to 𝑂(β„Žβˆ’1 β„“ ) splits can be used without increasing the path cost significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' If 𝑀ℓ splits are used, then the standard splitting variance analysis gives V[οΏ½π‘Œβ„“] = V οΏ½ E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 | {Ξ”π‘Šπ‘›}𝑛<𝑁 ] οΏ½ + π‘€βˆ’1 β„“ E οΏ½ V[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 | {Ξ”π‘Šπ‘›}𝑛<𝑁 ] οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' As discussed previously V οΏ½ E[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 | {Ξ”π‘Šπ‘›}𝑛<𝑁 ] οΏ½ = 𝑂(β„Ž3/2 β„“ ), and similarly it can be argued that E οΏ½ V[ οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 | {Ξ”π‘Šπ‘›}𝑛<𝑁 ] οΏ½ = 𝑂(β„Žβ„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Therefore choosing 𝑀ℓ to lie between 𝑂(β„Žβˆ’1 β„“ ) and 𝑂(β„Žβˆ’1/2 β„“ ) ensures the benefits of the splitting are obtained without significantly increasing the computational cost per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 St Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 4 Path splitting in final timestep to estimate conditional expectation MLMC techniques for discontinuous functions 11 As an additional bonus, one can use the more accurate Milstein discretisation for the final timestep, instead of switching to the Euler-Maruyama discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Burgos [9, 10] gives more details of the analysis, and also used the same approach for pathwise sensitivity analysis for a variety of financial options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles & Bernal [16] also used splitting for Feynman-Kac functionals arising for stopped diffusions, SDE simulations which terminate when the solution path leaves a prescribed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The issue here is that when a fine path exits, there is an 𝑂(β„Ž1/2 β„“ ) probability that the corresponding coarse path does not leave until much later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This is addressed by estimating a conditional expectation by splitting the coarse path into 𝑂(β„Žβˆ’1/2 β„“ ) independent sub-simulations which continue until each of them leaves the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 𝑉ℓ is improved from 𝑂(β„Ž1/2 β„“ ) to approximately 𝑂(β„Žβ„“) without a significant increase in the cost per sample, and finally the MLMC complexity achieved is 𝑂(πœ€βˆ’2| log πœ€|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' None of the three methods introduced so far (conditional expectation, change of measure, splitting) helps when using the Euler-Maruyama discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For this, a new method has recently been developed by Giles & Haji-Ali [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' It again uses splitting, but inspired by the simulation of branching diffusions, it considers splits at multiple deterministic times, as illustrated in Figure 5 which shows the logical structure of a set of split paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Here we are considering a simulation on the unit time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' A single pair of fine/coarse paths is calculated up to time 𝑑 = 1/2, with the number of fine timesteps being 1 2 β„Žβˆ’1 β„“ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This simulation is then split into two separate independent simulations up to time 𝑑 = 3/4, with the two simulations between them accounting for an additional 1 2 β„Žβˆ’1 β„“ fine timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' There are further splits at 𝑑 = 3/4, then at 𝑑 = 7/8, and so on, with the final split when there is just one coarse timestep left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The total number of fine timesteps simulated is 𝑂(β„Žβˆ’1 β„“ | log β„Žβ„“|) so the computa- tional cost is only slightly increased compared to the original method with a single pair of fine/coarse paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' οΏ½π‘Œβ„“ is defined to be the average of the values οΏ½π‘ƒβ„“βˆ’οΏ½π‘ƒβ„“βˆ’1 for each of the final paths, and it can be proved that its variance is 𝑂(β„Žβ„“), the same 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='8 1 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 5 Repeated path splitting to estimate conditional expectation 12 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles asymptotic order of convergence as for Lipschitz payoff functions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The kurtosis is also improved, so this technique fully addresses the challenge of using MLMC with the Euler-Maruyama discretisation to estimate digital option values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 7 Adaptive sampling We return now to the challenge of estimating the nested expectation E [ 𝐻 (E[𝑍|𝑋]) ] and we note that we only need an accurate estimate of the inner conditional expecta- tion E[𝑍|𝑋] when it is near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This observation is the basis for the development of adaptive sampling by Broadie, Du & Moallemi [7] within a standard Monte Carlo procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This was then extended to adaptive sampling combined with MLMC by Giles & Haji-Ali [19] by defining the number of inner samples 𝑀ℓ on level β„“ to be 𝑀ℓ = 2ℓ𝑀0 inner samples when |E[𝑍|𝑋]| ≫ √︁ V[𝑍|𝑋]/(2ℓ𝑀0) This is the smallest number of samples used on level β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' √︁ V[𝑍|𝑋]/(2ℓ𝑀0) is the standard deviation of the Monte Carlo estimate for E[𝑍|𝑋], so the inequality means that this number of samples is sufficient to be very sure that the estimate has the correct sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 𝑀ℓ = 4ℓ𝑀0 inner samples when |E[𝑍|𝑋]| = 𝑂( √︁ V[𝑍|𝑋]/(4ℓ𝑀0)) This is the maximum number of samples used on level β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In this case, the estimate of E[𝑍|𝑋] may have the incorrect sign, but this will only happen when |E[𝑍|𝑋]| = 𝑂(2βˆ’β„“) which occurs with probability 𝑂(2βˆ’β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Likewise, the total cost of the higher number of samples in this region is 𝑂(2βˆ’β„“ Γ— 4β„“) = 𝑂(2β„“), so it does not significantly increase the overall average cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 2ℓ𝑀0 < 𝑀ℓ < 4ℓ𝑀0 for intermediate values In this region the number of samples is chosen to be very sure that the estimate of E[𝑍|𝑋] has the correct sign, and at the same time the total cost is 𝑂(2β„“).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Overall, this adaptive sampling approach leads to 𝐢ℓ ∼ 2β„“, 𝑉ℓ ∼ 2βˆ’β„“ and hence a complexity of roughly 𝑂(πœ€βˆ’2) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' However, the kurtosis is 𝑂(2β„“) since only an 𝑂(2βˆ’β„“) fraction of the outer samples give non-zero values for οΏ½π‘Œβ„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Haji-Ali, Spence & Teckentrup [27] have further extended this to estimate quan- tities of the form P[𝐺 ∈ Ξ©] ≑ E[1𝐺∈Ω] where 𝐺 is a 𝑑-dimensional random variable which cannot be sampled directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In their paper they consider in particular the two challenges in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In the context of the digital option with the Euler-Maruyama discretisation on the unit time interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' the adaptive sampling varies the timestep used on level β„“ so that β„Žβ„“ = 2βˆ’β„“ when |�𝑆ℓ βˆ’ 𝐾| is large compared to the strong error in the path approx- imation β„Žβ„“ = 4βˆ’β„“ when |�𝑆ℓ βˆ’ 𝐾| is of the same order as the strong error MLMC techniques for discontinuous functions 13 2βˆ’β„“ < β„Žβ„“ < 4βˆ’β„“ for intermediate values A Brownian bridge construction is used when the timestep needs to be refined as part of the adaptation procedure from its initial value β„Žβ„“ = 2βˆ’β„“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The adaptation again leads to 𝐢ℓ ∼ 2β„“, 𝑉ℓ ∼ 2βˆ’β„“ and hence a complexity of roughly 𝑂(πœ€βˆ’2), but there is again a high kurtosis [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' In earlier research, Elfverson, Hellman & Malqvist [11] considered estimation of E[𝐻(𝑋)] where 𝑋 cannot be sampled exactly but there is a sequence of approxi- mations 𝑋′ 0, 𝑋′ 1, 𝑋′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 𝑋 of increasing accuracy and increasing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Motivated by PDE applications with a well-behaved truncation error so that there are uniform geometric bounds on |𝑋′ 𝑗 βˆ’ 𝑋|, level β„“ in their method uses �𝑋ℓ = 𝑋′ 𝑗, 𝑗 = min{β„“, min 𝑗 : | �𝑋′ 𝑗 βˆ’ 𝑋| < |𝑋|} and achieves similarly good MLMC benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' This idea is essentially the same as in the work of Haji-Ali et al but requiring a uniform bound on |𝑋′ π‘—βˆ’π‘‹| is significantly more restrictive than the bounds on E[ |𝑋′ π‘—βˆ’π‘‹|π‘ž] for some π‘ž>2 required by Haji-Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' A final comment is that the analysis of Haji-Ali, Spence & Teckentrup can be gen- eralised to a product of an indicator function and a Lipschitz function, E[1𝐺∈Ω 𝑓 (𝑆)], and so can handle barrier options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Furthermore, Haji-Ali & Spence have extended the adaptive sampling methodology to an extremely challenging triply-nested expec- tation which arises in mathematical finance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' By incorporating the randomised MLMC treatment of Rhee & Glynn [32] to handle the time discretisation of the underlying SDEs as well as the sampling for the inner conditional expectations, they achieve an overall complexity of approximately 𝑂(πœ€βˆ’2) which is very impressive for such a difficult application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 10 5 0 5 10 E[Z|X] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 6 Error distributions for two conditional expectations with i) few samples being needed to ensure the correct sign (left), and ii) many samples being insufficient to ensure the correct sign (centre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The blue line represents the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' 14 Michael B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Giles 8 Conclusions It is worth repeating that in most MLMC applications the output quantity of interest is a Lipschitz function of the intermediate simulation quantities, so good strong convergence for the intermediate quantities leads automatically to a good rate of convergence of the MLMC variance 𝑉ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' For those applications in which the function is discontinuous, this article shows there is an extensive literature with a variety of different approaches to improve the MLMC variance and try to recover the optimal 𝑂(πœ€βˆ’2) complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' It is notable that many of these methods have adapted ideas from Monte Carlo sensitivity analysis which also has problems with discontinuous functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' It is hoped that this survey will assist future researchers facing similar challenges in other new application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Acknowledgements This paper is based on research with many students, postdocs and other collaborators and I am grateful to all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' Funding for the research has been provided by the UK Engineering and Physical Sciences Research Council through grants EP/E031455/1, EP/H05183X/1 and EP/P020720/2 as well as the Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' The paper was written while visiting the Oden Institute at UT Austin, and I thank my hosts for their warm hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE1T4oBgHgl3EQfDwOw/content/2301.02882v1.pdf'} +page_content=' N.' metadata={'source': 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