Error expansion for the discretization of Backward Stochastic Differential Equations

Details Error expansion for the discretization of Backward Stochastic Differential Equations Error expansion for the discretization of Backward Stochastic Differential Equations

2006-02-22

arxiv.org/abs/math/0602503v1

Mathematics

Probability

27 pages

Scientific paper

We study the error induced by the time discretization of a 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 ($p \geq 1$) are of order $N^{-1/2}$ (this generalizes the results by Zhang 2004). Secondly, an error expansion is derived: surprisingly, the first term is proportional to $X^N-X$ while residual terms are of order $N^{-1}$.

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