Article2009

Error estimates of the $\theta$-scheme for backward stochastic differentialequations

58 citations0/0 files storedDiscrete and Continuous Dynamical Systems - B

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Stochastic processes and financial applications Numerical methods for differential equations Fuzzy Systems and Optimization

Abstract

In this paper, we study the error estimate of the $\theta$-schemefor the backward stochastic differential equation$y_t=\varphi(W_T)+\int_t^Tf(s,y_s)ds-\int_t^Tz_sdW_s$. We show thatthis scheme is of first-order convergence in $y$ for general$\theta$. In particular, for the case of $\theta=\frac{1}{2}$ (i.e.,the Crank-Nicolson scheme), we prove that this scheme is ofsecond-order convergence in $y$ and first-order in $z$. Somenumerical examples are also given to validate our theoreticalresults.

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References

2016Numerical solution of stochastic differential equations 1990Adapted solution of a backward stochastic differential equation 1997Backward Stochastic Differential Equations in Finance 1990A General Stochastic Maximum Principle for Optimal Control Problems 1994Solving forward-backward stochastic differential equations explicitly — a four step scheme 2007Forward-Backward Stochastic Differential Equations and their Applications 1991Probabilistic interpretation for systems of quasilinear parabolic partial differential equations 2004Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations

Cited by

2012A generalized $\theta$-scheme for solving backward stochastic differential equations 2013A Numerical Method and Its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations 2013A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations 2017Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs 2023Numerical methods for backward stochastic differential equations: A survey

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