On Fractional Brownian Motion with Application to Risk Sensitive

Abstract

This thesis expands upon Pontryagin s stochastic maximum principle to accommodate systems modeled by fractional Brownian motion. we present two research topics. The rst centers on an optimal control problem wherein the state equation is driven by fractional Brownian motion, and the cost functional follows a risk-neutral type. Ini- tially, we present the optimal control problem and its underlying dynamics, followed by the convex perturbation method in which the set of admissible controls is con- vex. Subsequently, we establish both optimality conditions for this model. Finally, we demonstrate our ndings through a linear quadratic problem, solving the associated Riccati type equation. The second topic focuses on characterizing optimal control prob- lems within a risk-sensitive framework. The system dynamics are de ned using only the backward stochastic di¤erential equations. However, the performance criterion is distinct; instead of directly minimizing costs, we aim to minimize a convex disutility function of the cost. As an initial step, we elucidate the relationship between risk- neutral and risk-sensitive loss functionals. Next, we establish the equivalence between expected exponential utility and quadratic backward stochastic di¤erential equations. Further, we reformulate the risk-sensitive problem into a standard risk-neutral one by introducing an auxiliary term and demonstrate the determination of the adjoint equa- tion. Thus, we derive the stochastic maximum principle using a standard application of risk-neutral results. Finally, we apply these concepts to a control problem with linear quadratic risk sensitivity.