A Donsker delta functional approach to optimal insider control and applications to finance
Journal article, Peer reviewed
Accepted version
Date
2015Metadata
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Original version
Communications in Mathematics and Statistics. 2015, 3 (3), 365-421. 10.1007/s40304-015-0065-yAbstract
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.