We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida-Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.

This paper deals with optimal combined singular and regular control for stochastic Volterra integral equations, where the solution X u,ξ (t) = X(t) is given byHere ξ denotes the singular control and u denotes the regular control.Unless otherwise stated, b a h(s)dξ(s) means [a,b] h(s)dξ(s). Such systems may for example be used to model for harvesting of populations with memory, where X(t) represents the population density at time t, and the singular control process ξ represents the harvesting effort rate. The total income from the harvesting is represented byfor given functions f 0 , f 1 and g, where T > 0 is a constant denoting the terminal time of the harvesting.

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Our main concern in this paper is to study mixed regular-singular control problems, where the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state process, as well as the running and final costs are random functions, so as the state process is no longer a Markov process. Our main result is to derive necessary conditions for optimality, also known as the Pontriagin stochastic maximum principle by using Malliavin calculus techniques. The adjoint process, which plays a key role in the stochastic maximum principle, is given by means of the Malliavin derivatives of the optimal state process.

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution X u,ξ (t) = X(t) is given by