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

Abstract

We study 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 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 Itˆo-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.