Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives

Abstract

In this article we consider a stochastic optimal control problem where
\nthe dynamics of the state process, X(t), is a controlled stochastic differential
\nequation with jumps, delay and noisy memory. The term noisy
\nmemory is, to the best of our knowledge, new. By this we mean that the
\ndynamics of X(t) depend on R t
\nt−δ X(s)dB(s) (where B(t) is a Brownian
\nmotion). Hence, the dependence is noisy because of the Brownian motion,
\nand it involves memory due to the influence from the previous values of
\nthe state process.
\nWe derive necessary and sufficient maximum principles for this stochastic
\ncontrol problem in two different ways, resulting in two sets of maximum
\nprinciples. The first set of maximum principles is derived using Malliavin
\ncalculus techniques, while the second set comes from reduction to a discrete
\ndelay optimal control problem, and application of previously known
\nresults by Øksendal, Sulem and Zhang. The maximum principles also
\napply to the case where the controller has only partial information, in the
\nsense that the admissible controls are adapted to a sub-σ-algebra of the
\nnatural filtration.