Feedback optimal control for stochastic Volterra equations with completely monotone kernels

Fulvia Confortola; Elisa Mastrogiacomo

doi:10.3934/mcrf.2015.5.191

Article Contents

2015, Volume 5, Issue 2: 191-235. Doi: 10.3934/mcrf.2015.5.191

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Feedback optimal control for stochastic Volterra equations with completely monotone kernels

Fulvia Confortola^1, and
Elisa Mastrogiacomo^2,

1.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133 Milano
2.
Dipartimento di statistica e metodi quantitativi, Universitá degli studi di Milano Bicocca, Piazza dell'Ateneo Nuovo, 1 - 20126, Milano

Received: August 31, 2013

Revised: July 31, 2014

Published: May 2015

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Abstract

In this paper we are concerned with a class of stochastic Volterra integro-differential problems with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup.
The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum $x$.

Keywords:

Stochastic Volterra integral equation,
backward stochastic differential equations,
optimal control,
HJB equation,
feedback law.

Mathematics Subject Classification: 45D05, 93E20, 60H30.

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