Singular optimal control of stochastic Volterra integral equations

TL;DR

This paper develops a framework for optimal control of stochastic Volterra integral equations with singular and regular controls, using Hida-Malliavin calculus to derive optimality conditions and applying results to harvesting models.

Contribution

It introduces a novel approach combining singular and regular controls for stochastic Volterra equations and derives new optimality conditions using Hida-Malliavin calculus.

Findings

Established necessary and sufficient conditions for optimal controls.

Derived a new class of backward stochastic Volterra integral equations.

Applied the theory to optimal harvesting with density-dependent prices.

Abstract

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution X^{u,\xi}(t)=X(t) is given by X(t) =\phi(t)+\int_{0}^{t}}b(t,s,X(s),u(s)) ds+\int_{0}^{t}\sigma(t,s,X(s),u(s))dB(s) +\int _{0}^{t}\int_{0}^{t}h(t,s) d\xi(s). Here dB(s) denotes the Brownian motion It\^o type differential and \xi denotes the singular control (singular in time t with respect to Lebesgue measure) and u denotes the regular control (absolutely continuous with respect to Lebesgue measure). Such systems may for example be used to model harvesting of populations with memory, where X(t) represents the population density at time t, and the singular control process \xi represents the harvesting effort rate. The total income from the harvesting is represented by J(u,\xi) =E[\int _{0}^{T}\int_{0}^{T} f_{0}(t,X(t),u(t))dt+\int _{0}^{T}\int_{0}^{T} f_{1}(t,X(t))d\xi(t)+g(X(T))], for given functions f_{0},f_{1} and g, where T>0 is a constant denoting the terminal time of the harvesting. Note that it is important to allow the controls to be singular, because in some cases the optimal controls are of this type. Using Hida-Malliavin calculus, we prove sufficient conditions and necessary conditions of optimality of controls. As a consequence, we obtain a new type of backward stochastic Volterra integral equations with singular drift. Finally, to illustrate our results, we apply them to discuss optimal harvesting problems with possibly density dependent prices.