Error expansion for the discretization of backward stochastic differential equations

We study the error induced by the time discretization of decoupled forward-backward stochastic differential equations (X,Y,Z). The forward component X is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme XN with N time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (YN-Y,ZN-Z) measured in the strong Lp-sense (p>=1) are of order N-1/2 (this generalizes the results by Zhang [J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability 14 (1) (2004) 459-488]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to XN-X while residual terms are of order N-1.

[1]  MA BYJIN,et al.  REPRESENTATION THEOREMS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS , 2002 .

[2]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[3]  Rémi Munos,et al.  Sensitivity Analysis Using It[o-circumflex]--Malliavin Calculus and Martingales, and Application to Stochastic Optimal Control , 2005, SIAM J. Control. Optim..

[4]  É. Pardoux.  Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Semilinear Parabolic and Elliptic PDEs of Second Order , 1998 .

[5]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[6]  B. Bouchard,et al.  Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , 2004 .

[7]  E. Gobet,et al.  Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations , 2006 .

[8]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[9]  Denis Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996, Monte Carlo Methods Appl..

[10]  Jianfeng Zhang.  A numerical scheme for BSDEs , 2004 .

[11]  D. Duffie,et al.  Mean-variance hedging in continuous time , 1991 .

[12]  J. Ma,et al.  Forward-Backward Stochastic Differential Equations and their Applications , 2007 .

[13]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[14]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[15]  D. Chevance,et al.  Numerical Methods in Finance: Numerical Methods for Backward Stochastic Differential Equations , 1997 .

[16]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

[17]  Arturo Kohatsu-Higa,et al.  Variance Reduction Methods for Simulation of Densities on Wiener Space , 2002, SIAM J. Numer. Anal..

[18]  Philip Protter,et al.  Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's , 1991 .

[19]  E. Gobet,et al.  A regression-based Monte Carlo method to solve backward stochastic differential equations , 2005, math/0508491.

[20]  P. Protter.  Stochastic integration and differential equations , 1990 .

[21]  S. Menozzi,et al.  A Forward-Backward Stochastic Algorithm For Quasi-Linear PDEs , 2006, math/0603250.

[22]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[23]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .