A numerical scheme for BSDEs
The author considers the decoupled system of forward-backward stochastic differential equations \[ \begin{aligned} X_t&=x + \int_0^t b(s,X_s) \,ds + \int_0^t \sigma(s,X_s)\,dW_s,\\ Y_t&= \Phi(X)+\int_t^Tf(s,X_s,Y_s,Z_s)\,ds-\int_t^TZ_s\,dW_s, \end{aligned} \tag{1} \] where \(b,\,\sigma\) and \(f\) are deterministic functions and \(W\) is a standard Brownian motion. The deterministic functional \(\Phi\) is assumed to be a so-called \(L^\infty\)-Lipschitz or a \(L^1\)-Lipschitz functional or takes the form \(g(X_T)\). Based on results of \textit{J. Ma} and \textit{J. Zhang} [Probab. Theory Relat. Fields 122, No. 2, 163--190 (2002; Zbl 1014.60060)] he proves \(L^2\)-regularity of the martingale integrand \(Z\). A numerical scheme is proposed, consisting of the Euler method for the forward diffusion \(X\) and an approximation via step processes for the pair \((Y,Z)\). Convergence in the mean-square sense of the method is proved and convergence rates are obtained. The latter depend in particular on the above stated smoothness properties of \(\Phi\).
backward stochastic differential equations
scientific article; zbMATH DE number 2072705