Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations

2022/04/18 by Martin Hutzenthaler, Hutzenthaler, Martin, Tuan A. Nguyen +1
Economics, Econometrics and Finance · Mathematics · #65C30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Primary 60H30 #Probability (math.PR) #Secondary 65C05 #Statistical Methods and Inference #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2204.08511

openalex publication_date 2022/04/18 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28

Abstract

Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE solution. Recently, Hutzenthaler et al. (arXiv:2108.10602) introduced a new approximation method for BSDEs whose forward diffusion is Brownian motion and proved that this method converges with essentially optimal rate without suffering from the curse of dimensionality. The central object of this article is to extend this result to general forward diffusions. The main challenge is that we need to establish convergence in temporal-spatial Hölder norms since the forward diffusion cannot be sampled exactly in general.

Related

• Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations
• Multilevel approximation of backward stochastic differential equations
• A Numerical Scheme For High-dimensional Backward Stochastic Differential Equation Based On Modified Multi-level Picard Iteration
• A forward scheme with machine learning for forward-backward SDEs with jumps by decoupling jumps
• Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion