Error expansion for the discretization of backward stochastic differential equations

Emmanuel Gobet; Céline Labart

doi:10.1016/j.spa.2006.10.007

Error expansion for the discretization of backward stochastic differential equations

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Abstract

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 X^N with N time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (Y^N - Y, Z^N - Z) measured in the strong L_p-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 X^N - X while residual terms are of order N^{- 1}.

Original language	English
Pages (from-to)	803-829
Number of pages	27
Journal	Stochastic Processes and their Applications
Volume	117
Issue number	7
DOIs	https://doi.org/10.1016/j.spa.2006.10.007
Publication status	Published - 1 Jul 2007
Externally published	Yes

Keywords

Backward stochastic differential equation
Discretization scheme
Malliavin calculus
Semi-linear parabolic PDE

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10.1016/j.spa.2006.10.007

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title = "Error expansion for the discretization of backward stochastic differential equations",

abstract = "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.",

keywords = "Backward stochastic differential equation, Discretization scheme, Malliavin calculus, Semi-linear parabolic PDE",

author = "Emmanuel Gobet and C{\'e}line Labart",

year = "2007",

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doi = "10.1016/j.spa.2006.10.007",

language = "English",

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journal = "Stochastic Processes and their Applications",

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TY - JOUR

T1 - Error expansion for the discretization of backward stochastic differential equations

AU - Gobet, Emmanuel

AU - Labart, Céline

PY - 2007/7/1

Y1 - 2007/7/1

N2 - 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.

AB - 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.

KW - Backward stochastic differential equation

KW - Discretization scheme

KW - Malliavin calculus

KW - Semi-linear parabolic PDE

U2 - 10.1016/j.spa.2006.10.007

DO - 10.1016/j.spa.2006.10.007

M3 - Article

AN - SCOPUS:34248575242

SN - 0304-4149

VL - 117

SP - 803

EP - 829

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 7

ER -

Error expansion for the discretization of backward stochastic differential equations

Abstract

Keywords

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