Backward Stochastic Differential Equation Driven by Fractional Brownian Motion

Abstract

We study general linear and nonlinear backward stochastic differential equations driven by fractional Brownian motions. The existence and uniqueness of the solutions are obtained under some mild assumptions. In the nonlinear case we obtain an inequality of the type similar to in the classical backward stochastic differential equations. This leads to a fixed point principle. An important tool is the quasi-conditional expectation introduced in [Y. Hu and B. Øksendal, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), pp. 1–32]. We also give a detailed study on this new “expectation.”