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

TL;DR

This paper develops a novel mathematical framework using anticipative white noise analysis and Donsker delta functionals to solve optimal insider control problems, with applications to finance involving complex stochastic systems.

Contribution

It introduces a new approach combining white noise analysis and Donsker delta functionals to handle insider control problems outside semimartingale frameworks.

Findings

Derived maximum principles for insider control systems.

Obtained explicit solutions for insider portfolio optimization.

Applied methods to financial models with Itô-Lévy processes.

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.