Computing anticipatory property in stochastic differential systems

This article concerns computing anticipatory property of stochastic systems, which are characterized by forward and/or backward stochastic differential equations. Since a Pardoux-Peng solution of a Backward Stochastic Differential Equation (BSDE) is required adaptedness to the natural filtration of a standard Brownian motion, it is nonanticipatory in the sense of stochastic calculus. However, the BSDE is solved backwardly with a terminal state and the solution will generally depend on the future state. Hence a system described by Forward and Backward Stochastic Differential Equations (FBSDEs) will be shown to be a strong computing anticipatory system. A stochastic system described by a numerical scheme also has the same property.