Linear-quadratic fractional Gaussian control

This is the published version, also available here: http://dx.doi.org/10.1137/120877283.In this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean, square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved. An optimal control is given explicitly as the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and the prediction of the response of a system to the future noise process. The optimal cost is also given. The special case of a noise process that is an arbitrary standard fractional Brownian motion is noted explicitly with an explicit expression for the prediction of the future response of a system to the noise process that is used the optimal control

About the linear-quadratic regulator problem under a fractional Brownian perturbation and complete observation

In this report we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion

Linear-quadratic control for stochastic equations in a Hilbert space with a fractional Brownian motion

This is the published version, also available here: http://dx.doi.org/10.1137/110831416.A linear-quadratic control problem with a finite time horizon for some infinite-dimensional controlled stochastic differential equations driven by a fractional Gaussian noise is formulated and solved. The feedback form of the optimal control and the optimal cost are given explicitly. The optimal control is the sum of the well-known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise. The covariance of the noise as well as the control operator in the system equation can in general be unbounded, so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain. Some examples of controlled stochastic partial differential equations are given

On the maximum principle for optimal control problems of stochastic Volterra integral equations with delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.Comment: 28 page

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of {\em covariance measure structure} for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion.Comment: 50 page

Rough volatility and portfolio optimisation under small transaction costs

The first chapter of the thesis presents the study of the linear-quadratic ergodic control problem of fractional Brownian motion. Ergodic control problems arise naturally in the context of small cost asymptotic expansion of utility maximisation problems with frictions. The optimal solution to the ergodic control problem is derived through the use of an infinite dimensional Markovian representation of fractional Brownian motion as a superposition of Ornstein-Uhlenbeck processes. This solution then allows to compute explicit formulas for the minimised objective value through the variance of the stationary distribution of the Ornstein-Uhlenbeck processes. Building on the first chapter, the second chapter of the thesis presents the main result. This is motivated by the problem an agent faces when trying to minimise her utility loss in the presence of quadratic trading costs in a rough volatility model. Minimising the utility loss amounts to studying a tracking problem of a target that depends on the rough volatility process. This tracking problem is minimised at leading order by an asymptotically optimal strategy that is closely linked to the ergodic control problem of fractional Brownian motion. This asymptotically optimal strategy is explicitly derived. Moreover, the leading order of the small cost expansion is shown to depend only on the roughest part of the considered target. It therefore depends on the Hurst parameter. The third chapter is devoted to a numerical analysis of the utility loss studied in the second chapter. For this, we compare the utility loss in a rough volatility model to a semimartingale stochastic volatility model. The parameter values for both models are fitted to match frictionless utility for realistic values. By applying the result obtained in the second chapter of the thesis, the difference between leading order of utility loss can be explicitly compared

Management of the orbital angular momentum of vortex beams in a quadratic nonlinear interaction

Light intensity control of the orbital angular momentum of the fundamental beam in a quadratic nonlinear process is theoretically and numerically presented. In particular we analyzed a seeded second harmonic generation process in presence of orbital angular momentum of the interacting beams due both to on axis and off axis optical vortices. Examples are proposed and discussed