Numerical method for backward stochastic differential equations
Let \(W\) be a \(d\)-dimensional Brownian motion. The authors develop a new method of approximating solutions \(Y\) of the multidimensional backward stochastic differential equation (BSDE) \[ dY_t= -f(t, Y_t)dt+ Z_t dW_t,\quad t\in [0,T], \] with a continuous driver \(f\) which is Lipschtz in the \(y\)-variable and independent of \(z\). As a consequence the interest in numerical treatments of such equations in finance, several numerical approximating methods for BSDEs have already been developed, cf. \textit{J. Douglas jun.}, \textit{J. Ma} and \textit{P. Protter} [Ann. Appl. Probab. 6, No. 3, 940-968 (1996; Zbl 0861.65131)], \textit{V. Bally} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 177-191 (1997; Zbl 0889.60068)], \textit{D. Chevance} [in: Numerical methods in finance, 232-244 (1997; Zbl 0898.90031)] and \textit{V. Bally}, \textit{G. Pagès} and \textit{J. Printems} [Monte Carlo Methods Appl. 7, No. 1/2, 21-33 (2001)]. Another type of approximating BSDEs uses the discretization of the filtration [e.g. \textit{F. Coquet}, \textit{V. Mackevičius} and \textit{J. Mémin}, Stochastic Processes Appl. 75, No. 2, 235-248 (1998; Zbl 0932.60047)]. In their present paper the authors propose a discretization of the above BSDE where the Brownian motion is replaced by a simple random walk. Although the method of the authors uses ideas similar to those used in the above papers, in that they also approximate the Brownian motion by a discrete process, their main feature is that they do not need the convergence of the discretized filtrations to the original Brownian one in order to get the weak convergence in the Skorokhod topology of the solutions. Such a relaxation reduces the complexity in the construction of approximating solutions.
backward stochastic differential equations
scientific article; zbMATH DE number 1906190