Linear--Quadratic control for a class of stochastic Volterra equations: solvability and approximation

Abstract

We provide an exhaustive treatment of Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than $1/2$ as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state variables. Using a refined martingale verification argument combined with a squares completion technique, we prove that the value function is of linear quadratic form in these state variables with a linear optimal feedback control, depending on non-standard Banach space valued Riccati equations. Furthermore, we show that the value function of the stochastic Volterra optimization problem can be approximated by that of conventional finite dimensional Markovian Linear--Quadratic problems, which is of crucial importance for numerical implementation.

40 pages

References in corpus (2)

• Uniqueness for Volterra-type stochastic integral equations
• Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise

Cited by in corpus (4)

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• Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals
• The Laplace transform of the integrated Volterra Wishart process
• Infinite horizon backward stochastic Volterra integral equations and discounted control problems