Error expansion for the discretization of backward stochastic differential equations

By Emmanuel Gobet, Céline Labart. 2006

109 citations recorded by the source corpus.

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.

Subfields: Demography, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/41ed86f79648abd2bd94d15bbc5e9eb472d6cc12

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Citations in the corpus

Second order discretization of backward SDEs and simulation with the cubature method (2014, methodology): “With additional smoothness assumptions imposed on the coefficients, Gobet and Labart [13] 3See Bender and Zhang [4] for an alternative approach based on Picard iterations. that the rate of convergence of the processes (8) is of order n−1.”
On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights (2010, methodology): “In contrast with the above literature, we concenrate on approximations of solutions of backward stochastic differential equations based on the Monte Carlo simulation of the underlying Brownian motion, thus continuing the studies of Bouchard and Touzi [6], Zhang [18] and Gobet, Lemor and Warin [9].”
Numerical simulation of quadratic BSDEs (2013, unknown): “5, that is to say the minimal rate proved in this paper, and 1 the general optimal rate for the Euler scheme; see, for example, [11, 22].”
Error estimates of the $\theta$-scheme for backward stochastic differentialequations (2009, background): “generalized Zhang’s result in the strong L-sense (p ≥ 1) [11] and proposed a numerical scheme based on iterative regressions on function bases [12].”
RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS (2014, methodology): “These schemes are based on the well-known Runge–Kutta methods for ODEs and include new high order schemes as well as existing low order schemes such as the classical extension of the Euler scheme to BSDEs; see, for example, [1, 2, 4, 6].”
Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition (2011, unknown): “In this situation, authors of [15] show that the discretization error for the BSDE is on the same order than the discretization error for the SDE if we assume extra smoothness assumptions on b, σ, g and f .”