Error expansion for the discretization of backward stochastic differential equations
Copyright policy )Error expansion for the discretization of backward stochastic differential equations
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 $N^{-1}$.
27 pages
- École Polytechnique France
- Grenoble Alpes University France
Statistics and Probability, Semi-linear parabolic PDE, discretization scheme, Stochastic calculus of variations and the Malliavin calculus, Applied Mathematics, Malliavin calculus, Probability (math.PR), backward stochastic differential equation, Central limit and other weak theorems, Error analysis and interval analysis, Discretization scheme, 510, Backward stochastic differential equation, Stochastic ordinary differential equations (aspects of stochastic analysis), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60H07, 60F05, 60H10, 65G99, semi-linear parabolic PDE, Modelling and Simulation, MSC 60H07 60F05 60H10 65G99, FOS: Mathematics, Mathematics - Probability
Statistics and Probability, Semi-linear parabolic PDE, discretization scheme, Stochastic calculus of variations and the Malliavin calculus, Applied Mathematics, Malliavin calculus, Probability (math.PR), backward stochastic differential equation, Central limit and other weak theorems, Error analysis and interval analysis, Discretization scheme, 510, Backward stochastic differential equation, Stochastic ordinary differential equations (aspects of stochastic analysis), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60H07, 60F05, 60H10, 65G99, semi-linear parabolic PDE, Modelling and Simulation, MSC 60H07 60F05 60H10 65G99, FOS: Mathematics, Mathematics - Probability
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