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 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 nonstandard 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.
The work of Huyên Pham is supported by FiME (Finance for Energy Market Research Centre) and the “Finance et Développement Durable—Approches Quantitatives” EDF—CACIB Chair.
"Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation." Ann. Appl. Probab. 31 (5) 2244 - 2274, October 2021. https://doi.org/10.1214/20-AAP1645