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October 2021 Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation
Eduardo Abi Jaber, Enzo Miller, Huyên Pham
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Ann. Appl. Probab. 31(5): 2244-2274 (October 2021). DOI: 10.1214/20-AAP1645

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 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.

Funding Statement

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.

Citation

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Eduardo Abi Jaber. Enzo Miller. Huyên Pham. "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

Information

Received: 1 November 2019; Revised: 1 June 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1645

Subjects:
Primary: 93E20
Secondary: 49N10, 60G22, 60H20

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 5 • October 2021
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