Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs

Gobet, Emmanuel; López-Salas, José G.; Turkedjiev, Plamen; Vázquez, Carlos

doi:10.1137/16m106371x

Cited by 51 publications

(44 citation statements)

References 15 publications

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“…As pointed out in the introduction, many (time) discretization schemes and several (spacial) numerical approximation method of the solution of such as BSDE are proposed in the literature (we refer for example to [1,3,7,11,14,10,2,20]). Our aim in this section is to test the performances of our method to the numerical scheme proposed in [20].…”

Section: Approximation Of Bsdementioning

confidence: 99%

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Fiorin

Pagès²,

Sagna

2018

Methodol Comput Appl Probab

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We introduce a new approach to quantize the Euler scheme of an R d -valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast online quantization in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step t k is a cumulative of the marginal quantization error up to time t k . Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method.

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Section: Approximation Of Bsdementioning

confidence: 99%

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Fiorin

Pagès²,

Sagna

2018

Methodol Comput Appl Probab

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“…This is a considerable reduction of computational time, especially as the dimension q grows and correspondingly the dimension of the basis for approximating z i grows. Highlighting the computational time may be not the most important issues when solving BSDEs: in some situations (memory constraint in large dimension [19] or in parallel computing [17], calibration on small data [16]), the number of paths to be used is imposed and the objectives become to obtain the most efficient algorithm with a given Monte Carlo simulation effort. In that case, the use of ISMWDP type scheme for reducing statistical error is certainly advantageous.…”

Section: Concluding Remarks On the Numerical Experimentsmentioning

confidence: 99%

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

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“…Most of the above methods do not allow parallel computations because of the non-linearity of the equations. Recently in [21], the authors have proposed a novel approach inspired from regression Monte-Carlo schemes suitable for massive parallelization, with excellent results on GPUs. It takes advantage of local approximations of the regression function in different strata of the state space R d (partitioning estimates) and of a new stratified sampling scheme to guarantee that enough simulations are available in each stratum to compute accurately the regression functions.…”

Section: Introductionmentioning

confidence: 99%

Quasi-Regression Monte-Carlo Scheme for Semi-Linear PDEs and BSDEs with Large Scale Parallelization on GPUs

Gobet

López-Salas

Turkedjiev

et al. 2019

Arch Computat Methods Eng

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In this article we design a novel quasi-regression Monte Carlo algorithm in order to approximate the solution of discrete time backward stochastic differential equations (BSDEs), and we analyze the convergence of the proposed method. The algorithm also approximates the solution to the related semi-linear parabolic partial differential equation (PDE) obtained through the well known Feynman-Kac representation. For the sake of enriching the algorithm with high order convergence a weighted approximation of the solution is computed and appropriate conditions on the parameters of the method are inferred. With the challenge of tackling problems in high dimensions we propose suitable projections of the solution and efficient parallelizations of the algorithm taking advantage of powerful many core processors such as graphics processing units (GPUs).Assuming that u is smooth enough, the Itô formula leads toUsing the PDE (1.3) and noting u(T, X T ) = g(X T ), we get u(t, X t ) = g(X T )+ T t f (s, X s , u(s, X s ), (∇ x uσ)(s, X s ))ds− T t (∇ x uσ)(s, X s )dW s ,

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Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs

Cited by 51 publications

References 15 publications

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Quasi-Regression Monte-Carlo Scheme for Semi-Linear PDEs and BSDEs with Large Scale Parallelization on GPUs

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