Advances in Noise Modeling for Stochastic Systems in Optimal Control

Abstract

In this paper some noise models for stochastic control systems are described that differ from the well known model of Brownian motion. Some of the processes are Gaussian such as the family of fractional Brownian motions and other processes are non-Gaussian, especially Rosenblatt processes. These processes have a long range dependence property that can be described by a slow decay of the covariance process. A stochastic calculus for these processes exists that is more limited than the calculus for Brownian motion that is inherited from the martingale property but nevertheless is sufficient for addressing some stochastic control problems. In many physical situations the data demonstrates a long range dependence which can justify a choice from these non-Brownian processes. Explicit solutions of stochastic control problems with quadratic cost functionals having driving noise from the Rosenblatt processes are given.