The research focuses on stochastic analysis for systems with fractional Brownian motions with particular application to control problems. Since the available results for the solutions of stochastic differential equations driven by fractional Brownian motions are very limited, the investigation of explicit solutions of multidimensional bilinear equations and the existence and uniqueness of solutions of nonlinear equations is proposed. The explicit solutions for bilinear equations requires a combination of Lie theory and stochastic analysis and the explicit solutions provide an important method for solving various stochastic problems for bilinear equations. These stochastic differential equations should provide useful models for many physical phenomena. The stochastic optimal control of multidimensional linear systems driven by fractional Brownian motions and a quadratic cost functional for a finite time interval is proposed. Furthermore this controlled system with an ergodic cost is also proposed for study. In both cases the optimal control uses a prediction of the increments of a fractional Brownian motion. The ergodic control problem is the natural setting for an adaptive control problem for these linear systems. The adaptive control problem requires the identification of the unknown parameters of the linear system and the construction of a self-optimizing adaptive control. Some parameter identification schemes are proposed such as a weighted pseudo least squares algorithm to obtain strongly consistent estimators of the parameters.
The proposal describes the investigation of some stochastic models that use a family of stochastic processes called fractional Brownian motions which arose empirically in a model for the rainfall along the Nile River. The potential usage of these processes has been demonstrated for economic data, telecommunications, device noise, and medicine. To have useful stochastic models with fractional Brownian motions it is necessary to have information about the solutions of the equations. This is one goal of the proposal. Many stochastic models are also controlled and with a cost criterion an optimal control is sought. In the research some control problems for linear systems with a quadratic cost functional will be studied. Often some parameters of the system are unknown and it is also required to control the system. These problems require that the parameters are identified and a control is determined based on the estimates of the parameters. Such problems with controlled linear systems and fractional Brownian motion will be investigated.
