Numerical simulation of quadratic BSDEs

Abstract

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\frac{1}{2}-\epsilon$ for all $\epsilon >0$. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.

Published at http://dx.doi.org/10.1214/14-AAP1090 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• A regression-based Monte Carlo method to solve backward stochastic differential equations
• BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces
• Runge-Kutta schemes for backward stochastic differential equations
• Linear multi-step schemes for BSDEs

Cited by in corpus (20)

• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Deep splitting method for parabolic PDEs
• Learning Optimal Control via Forward and Backward Stochastic Differential Equations
• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression
• Numerical methods for backward stochastic differential equations: A survey
• Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs
• A primal-dual algorithm for BSDEs
• Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications
• Generalised multilevel Picard approximations
• Density analysis of BSDEs
• Actor-Critic Algorithm for High-dimensional Partial Differential Equations
• Multilevel approximation of backward stochastic differential equations
• Adapted time steps explicit scheme for monotone BSDEs
• A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs
• An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians
• Quantization-based approximation of reflected BSDEs with extended upper bounds for recursive quantization
• Full-Projection explicit FBSDE scheme for parabolic PDEs with superlinear nonlinearities
• Numerical stability analysis of the Euler scheme for BSDEs