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

Øksendal, Bernt; Mohammed, Salah-Eldin; Røse, Elin Engen; Dahl, Kristina Rognlien
Journal article, Peer reviewed
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URI
http://hdl.handle.net/11250/2466169
Date
2016
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  • Articles (FOR) [100]
  • Publikasjoner fra CRIStin (NHH) [243]
Original version
Journal of Functional Analysis. 2016, 271 (2), 289-329.   10.1016/j.jfa.2016.04.031
Abstract
In this article we consider a stochastic optimal control problem where

the dynamics of the state process, X(t), is a controlled stochastic differential

equation with jumps, delay and noisy memory. The term noisy

memory is, to the best of our knowledge, new. By this we mean that the

dynamics of X(t) depend on R t

t−δ X(s)dB(s) (where B(t) is a Brownian

motion). Hence, the dependence is noisy because of the Brownian motion,

and it involves memory due to the influence from the previous values of

the state process.

We derive necessary and sufficient maximum principles for this stochastic

control problem in two different ways, resulting in two sets of maximum

principles. The first set of maximum principles is derived using Malliavin

calculus techniques, while the second set comes from reduction to a discrete

delay optimal control problem, and application of previously known

results by Øksendal, Sulem and Zhang. The maximum principles also

apply to the case where the controller has only partial information, in the

sense that the admissible controls are adapted to a sub-σ-algebra of the

natural filtration.
Journal
Journal of Functional Analysis

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