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arXiv · 1504.02581

A Donsker delta functional approach to optimal insider control and applications to finance

Olfa Draouil·Bernt Øksendal
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Abstract

We study \emph{optimal insider control problems}, i.e. optimal control problems of stochastic systems where the controller at any time $t$ in addition to knowledge about the history of the system up to this time, also has additional information related to a \emph{future} value of the system. Since this puts the associated controlled systems outside the context of semimartingales, we apply anticipative white noise analysis, including forward integration and Hida-Malliavin calculus to study the problem. Combining this with Donsker delta functionals we transform the insider control problem into a classical (but parametrised) adapted control system, albeit with a non-classical performance functional. We establish a sufficient and a necessary maximum principle for such systems. Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by It\^ o-L\' evy processes. Finally, in the Appendix we give a brief survey of the concepts and results we need from the theory of white noise, forward integrals and Hida-Malliavin calculus.

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Olfa Draouil, Bernt Øksendal. 2015-04-10. A Donsker delta functional approach to optimal insider control and applications to finance. https://arxiv.org/abs/1504.02581

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