Numerical methods for backward stochastic differential equations: A survey
arXiv:2101.08936 · doi:10.1214/23-PS18
Abstract
Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping problems and gene expression. Most BSDEs cannot be solved analytically and thus numerical methods must be applied to approximate their solutions. There have been a variety of numerical methods proposed over the past few decades as well as many more currently being developed. For the most part, they exist in a complex and scattered manner with each requiring a variety of assumptions and conditions. The aim of the present work is thus to systematically survey various numerical methods for BSDEs, and in particular, compare and categorize them, for further developments and improvements. To achieve this goal, we focus primarily on the core features of each method based on an extensive collection of 333 references: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, to provide an up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and a useful comparison and categorization.
arXiv comment51 pages
References in corpus (55)
- DeepXDE: A deep learning library for solving differential equations
- DGM: A deep learning algorithm for solving partial differential equations
- Solving high-dimensional partial differential equations using deep learning
- Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
- Utility maximization in incomplete markets
- A regression-based Monte Carlo method to solve backward stochastic differential equations
- Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
- Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine 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
- A forward--backward stochastic algorithm for quasi-linear PDEs
- Convergence of the Deep BSDE Method for Coupled FBSDEs
- On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
- Deep splitting method for parabolic PDEs
- An overview on deep learning-based approximation methods for partial differential equations
- A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
- Time discretization and Markovian iteration for coupled FBSDEs
- Simulation of BSDEs by Wiener chaos expansion
- Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
- Propagation of chaos: a review of models, methods and applications. II. Applications
- Path regularity and explicit convergence rate for BSDE with truncated quadratic growth
- Numerical simulation of quadratic BSDEs
- A monotone scheme for high-dimensional fully nonlinear PDEs
- Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
- Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
- A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients
- Malliavin calculus for backward stochastic differential equations and application to numerical solutions
- Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression
- Arbitrage-Free XVA
- Runge-Kutta schemes for backward stochastic differential equations
- A cubature based algorithm to solve decoupled McKean-Vlasov Forward Backward Stochastic Differential Equations
- Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs
- Uniform error estimates for artificial neural network approximations for heat equations
- A primal-dual algorithm for BSDEs
- BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness
- Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
- A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver
- Discrete-time approximation of multidimensional BSDEs with oblique reflections
- On the homotopy analysis method for backward/forward-backward stochastic differential equations
- Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients
- Neural networks-based algorithms for stochastic control and PDEs in finance
- Stochastic grid bundling method for backward stochastic differential equations
- On the wavelets-based SWIFT method for backward stochastic differential equations
- Second-order numerical schemes for decoupled forward-backward stochastic differential equations with jumps
- Numerical aspects of shot noise representation of infinitely divisible laws and related processes
- A convolution method for numerical solution of backward stochastic differential equations
- Convergence Error Estimates of the Crank-Nicolson Scheme for Solving Decoupled FBSDEs
- Numerical approximations of McKean Anticipative Backward Stochastic Differential Equations arising in Initial Margin requirements
- Backward stochastic differential equation approach to modeling of gene expression
- Quasi-Regression Monte-Carlo scheme for semi-linear PDEs and BSDEs with large scale parallelization on GPUs
- Multistep schemes for solving backward stochastic differential equations on GPU
- Pricing Barrier Options with DeepBSDEs
- Approximations for adapted M-solutions of Type-II backward stochastic Volterra integral equations
- A numerical scheme for the quantile hedging problem
- An efficient third-order scheme for BSDEs based on nonequidistant difference scheme
- Numerical approximation of singular Forward-Backward SDEs