Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations

Abstract

In this paper, we study linear-quadratic control problems for stochastic Volterra integral equations with singular and non-convolution-type coefficients. The weighting matrices in the cost functional are not assumed to be non-negative definite. From a new viewpoint, we formulate a framework of causal feedback strategies. The existence and the uniqueness of a causal feedback optimal strategy are characterized by means of the corresponding Riccati-Volterra equation.

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

“…The results of Theorems 2.7 and 2.8 significantly improve various results for linear-quadratic (LQ) stochatic mean-field games that have appeared recently in the literature. In [22,23] a class of LQ stochastic game with state variables that satisfy Volterra integral equations was studied. The solution to the game is described in terms of a system of stochastic Volterra equations.…”

“…Solutions to Volterra LQ control problems can be characterised in some cases in terms of solutions to operator Riccati equations and to L 2 -valued BSDEs; see [2,Section 6]. In other cases they can be related to infinite dimensional Riccati equations, see [3,22,37] for single-player examples. Our derivation provides explicit operator solutions to such Riccati equations for the case of dynamics without control in the volatility.…”

Time‐inconsistent contract theory

“…1 We refer to the monograph Cvitanić and Zhang (2012) for a general framework that systematically surveys a great portion of the literature exploiting the maximum principle, in models driven by Brownian motion. 2 Ever since, several works have extended this approach, a probably incomplete list includes Shi et al (2015), Wang (2018), and Hamaguchi and Wang (2022). 3 The integrability for 𝑠 ∈ [0, 𝑇] fixed is clear.…”