Error expansion for the discretization of Backward Stochastic Differential Equations
نویسندگان
چکیده
We study the error induced by the time discretization of a decoupled forwardbackward 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 (Y N −Y,ZN −Z) measured in the strong Lp-sense (p ≥ 1) are of order N−1/2 (this generalizes the results by Zhang [20]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to XN − X while residual terms are of order N−1.
منابع مشابه
Stability of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type
This paper examines stability analysis of two classes of improved backward Euler methods, namely split-step $(theta, lambda)$-backward Euler (SSBE) and semi-implicit $(theta,lambda)$-Euler (SIE) methods, for nonlinear neutral stochastic delay differential equations (NSDDEs). It is proved that the SSBE method with $theta, lambdain(0,1]$ can recover the exponential mean-square stability with some...
متن کاملA Sparse-grid Method for Multi-dimensional Backward Stochastic Differential Equations
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hier...
متن کاملApproximate solution of the stochastic Volterra integral equations via expansion method
In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique give...
متن کاملConvergence of Legendre wavelet collocation method for solving nonlinear Stratonovich Volterra integral equations
In this paper, we apply Legendre wavelet collocation method to obtain the approximate solution of nonlinear Stratonovich Volterra integral equations. The main advantage of this method is that Legendre wavelet has orthogonality property and therefore coefficients of expansion are easily calculated. By using this method, the solution of nonlinear Stratonovich Volterra integral equation reduces to...
متن کاملApproximation of backward stochastic differential equations using Malliavin weights and least-squares regression
We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the Z-component by [18]. When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight...
متن کامل