Stochastic Control for Linear Systems Driven by Fractional Noises

Abstract

This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and studied. In the first two models, the state is scalar-valued and the control is taken as Markovian. Either the problems are completely solved based on a Riccati equation (for model 1, where the cost is a quadratic functional on state and control variables) or optimality is characterized (for model 2, where the cost is a power functional). The last control model under investigation is a general one, where the system involves the Stratonovich integral with respect to FBM, the state is multidimensional, and the admissible controls are not limited to being Markovian. A new Riccati-type equation, which is a backward stochastic differential equation involving both FBM and normal Brownian motion, is introduced. Optimal control and optimal value of the model are explicitly obtained based on the solution to this Riccati-type equation.

Mean-Field SDE Driven by a Fractional Brownian Motion and Related Stochastic Control Problem

“…Biagini et al [1] obtained a maximum principle for a stochastic control problem driven by an m-dimensional fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) m . For H ∈ (0, 1/2), Hu and Zhou [12] considered a linear stochastic optimal control problem and obtained a Riccati equation, a BSDE driven by the fractional Brownian motion and the underlying Brownian motion. Han et al [10] obtained a stochastic maximum principle for a control problem driven by a fractional Brownian motion with H > 1/2 and their adjoint equations is a linear BSDE again driven by the fractional Brownian motion and the underlying Brownian motion.…”

Linear-Quadratic Fractional Gaussian Control

“…Some special cases of a multidimensional system are described in [3] and [4]. Hu and Zhou [8] obtained an optimal control for a scalar bilinear system with a fractional Brownian motion for H ∈ (1/2, 1) but with the condition that the control is Markovian. In [2] the optimal control of a linear equation in a Hilbert space with a fractional Brownian motion having H ∈ ( 1 2 , 1) and with a quadratic cost functional is explicitly solved.…”

Optimal Control of a Stochastic Processing System Driven by a Fractional Brownian Motion Input

“…Usually, the controlled state process is a solution of a linear (semilinear in [8]) stochastic differential equation driven by FBM, and the control typically affects the drift term of the stochastic differential equation. In particular, the linear-quadratic regulator control problem is addressed in [16] and [17], and a stochastic maximum principle is developed and applied to several stochastic control problems in [3]. We refer the reader to [16] and to Chapter 9 of [4] for further examples of control problems in this setting.…”

“…In particular, the linear-quadratic regulator control problem is addressed in [16] and [17], and a stochastic maximum principle is developed and applied to several stochastic control problems in [3]. We refer the reader to [16] and to Chapter 9 of [4] for further examples of control problems in this setting. In contrast to the models considered in the above references, the model described here is motivated by queueing applications and involves processes with state constraints.…”

“…At the end of this section, we discuss an example of a queueing network which leads to our model. Our analysis relies on a coupling of the state process with its stationary version (see [20] and [37]) which enables us to address control problems in a non-Markovian setting, and our techniques are different from those employed in [3], [8], [16], and [17].…”