On the robustness of backward stochastic differential equations.
The backward stochastic differential equation driven by a Brownian motion \(W=\{W_t\} _{0\leq t\leq T}\), \[ Y_t= \xi + \int _t^T f(r,Y_r, Z_r)\,dr -\int _t^T Z_r\,dW_r,\quad 0\leq t\leq T, \] is considered, where the solution \((Y_t, Z_t)\) is supposed to be progressively measurable with respect to the filtration \(\{\mathcal F_t \}\) defined by the Brownian motion [for existence and uniqueness of such solution under suitable conditions see e.g.\ \textit{E. Pardoux} and \textit{S. G. Peng}, Syst. Control Lett. 14, 55--61 (1990; Zbl 0692.93064)]. The main result of the paper is ``robustness'' of solutions: If \((\xi ^n, f^n, W^n)\) converges to \((\xi , f,W)\) in an appropriate sense, the solutions of approximating equations converge to the solution of the limiting one as well. It should be noted that \(W^n\) is not assumed to have the predictable representation property. As a byproduct, the convergence of a ``Euler scheme'' for BSDE is obtained.
backward stochastic differential equation
stability of backward stochastic differential equations
scientific article; zbMATH DE number 2138939
scientific article