Stochastic Linear-Quadratic Control with State Dependent Fractional Brownian Noise and Stochastic Coefficients

Stochastic linear-quadratic control problems for bilinear equations with some stochastic coefficients and driven by fractional Brownian motions with the Hurst parameters H ϵ (½,1) are formulated and solved. The family of admissible controls is the collection of linear feedback controls. An optimal control in this family is not optimal in the admissible family of all adapted controls. However an optimal linear feedback control obtained in this paper can be relatively easily implemented whereas an optimal control in the family of adapted controls has a term that predicts the future noise so it is more difficult to implement. An optimal control is determined from the solution of a Riccati equation which differs from the Riccati equation for the linear-quadratic control problem with a Brownian motion replacing the noise term. Since the Riccati equation has some stochastic coefficients it is solved as a backward stochastic differential equation. Given the solution of the stochastic Riccati equation, a direct method is used to determine the optimal linear feedback control.