Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.
We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.
We consider a problem of optimal control of an infinite horizon system governed by forward-backward stochastic differential equations with delay. Sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived. We illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay.
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.
By a memory mean-field process we mean the solution X(·) of a stochastic mean-field equation involving not just the current state X(t) and its law L(X(t)) at time t, but also the state values X(s) and its law L(X(s)) at some previous times s < t. Our purpose is to study stochastic control problems of memory mean-field processes.• We consider the space M of measures on R with the norm || · || M introduced by Agram and Øksendal in [1], and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations.• We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-) advanced backward stochastic differential equations, one of them with values in the space of bounded linear functionals on path segment spaces.• As an application of our methods, we solve a memory mean-variance problem as well as a linear-quadratic problem of a memory process.MSC (2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.
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