A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for deep BSDE solver

DOI10.1016/j.jcp.2022.110956OpenAlexW3125111592WikidataQ114163372 ScholiaQ114163372MaRDI QIDQ2133701

Akihiko Takahashi, Toshihiro Yamada, Yoshifumi Tsuchida

Publication date: 5 May 2022

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2101.09890

zbMATH Keywords

asymptotic expansionbackward stochastic differential equationssemilinear partial differential equationsdeep learningcontrol variate methoddeep BSDE solver

Mathematics Subject Classification ID

Stochastic analysis (60Hxx) Actuarial science and mathematical finance (91Gxx) Probabilistic methods, stochastic differential equations (65Cxx)

Related Items (6)

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations ⋮ Control variate method for deep BSDE solver using weak approximation ⋮ Numerical methods for backward stochastic differential equations: a survey ⋮ Deep Weak Approximation of SDEs: A Spatial Approximation Scheme for Solving Kolmogorov Equations ⋮ Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus ⋮ Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations

Uses Software

• DGM
• Deep xVA solver

Cites Work

• Unnamed Item
• Unnamed Item
• A weak approximation with asymptotic expansion and multidimensional Malliavin weights
• Property and numerical simulation of the Ait-Sahalia-Rho model with nonlinear growth conditions
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Weak adversarial networks for high-dimensional partial differential equations
• Variational training of neural network approximations of solution maps for physical models
• A new computational scheme for computing Greeks by the asymptotic expansion approach
• An asymptotic expansion approach to pricing financial contingent claims
• Highly nonlinear model in finance and convergence of Monte Carlo simulations
• Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels
• On validity of the asymptotic expansion approach in contingent claim analysis
• Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions
• DGM: a deep learning algorithm for solving partial differential equations
• Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
• Solving high-dimensional eigenvalue problems using deep neural networks: a diffusion Monte Carlo like approach
• Acceleration of automatic differentiation of solutions to parabolic partial differential equations: a higher order discretization
• Solving many-electron Schrödinger equation using deep neural networks
• Convergence of the deep BSDE method for coupled FBSDEs
• A third-order weak approximation of multidimensional Itô stochastic differential equations
• On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
• Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs
• A control variate method for weak approximation of SDEs via discretization of numerical error of asymptotic expansion
• Stochastic calculus of variations in mathematical finance.
• The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims
• The Malliavin Calculus and Related Topics
• On Error Estimates for Asymptotic Expansions with Malliavin Weights: Application to Stochastic Volatility Model
• Backward Stochastic Differential Equations in Finance
• Asymptotic Expansion Approach in Finance
• An Arbitrary High Order Weak Approximation of SDE and Malliavin Monte Carlo: Analysis of Probability Distribution Functions
• An Asymptotic Expansion with Push-Down of Malliavin Weights
• Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels
• Solving high-dimensional partial differential equations using deep learning
• Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black--Scholes Partial Differential Equations
• Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
• Pricing Average and Spread Options Under Local-Stochastic Volatility Jump-Diffusion Models
• Second Order Discretization of Bismut--Elworthy--Li Formula: Application to Sensitivity Analysis
• Backward Stochastic Differential Equations

This page was built for publication: A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for deep BSDE solver