Singular optimal control of stochastic Volterra integral equations

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.