Some Stochastic Systems with a Fractional Brownian Motion and Applications to Control

In this paper some stochastic systems are considered that are described by stochastic differential equations with a fractional Brownian motion. The notion of a weak solution is introduced and obtained by a transformation of the measure for a fractional Brownian motion by a Radon-Nikodym derivative. This weak solution approach is used to solve a control problem for a controlled stochastic differential equation with a fractional Brownian motion and to verify the existence of an optimal control. An estimation problem for a stochastic signal observed with an additive fractional Brownian motion is formulated and solved. The conditional expectation which solves this problem is exhibited explicitly.