Malliavin calculus and optimal control of stochastic Volterra equations

TL;DR

This paper develops a Malliavin calculus-based maximum principle for optimal control of non-Markovian stochastic Volterra equations, including cases with partial information, and applies it to financial portfolio optimization.

Contribution

It introduces a novel maximum principle for stochastic Volterra equations using Malliavin calculus, extending control methods to non-Markovian systems with partial information.

Findings

Derived a sufficient maximum principle for stochastic Volterra equations.

Established a necessary maximum principle for such systems.

Applied the theory to solve an optimal portfolio problem with memory effects.

Abstract

Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore classical methods, like dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that by using {\em Malliavin calculus} it is possible to formulate a modified functional type of {\em 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 sufficient and a necessary 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.