Stochastic Control for Imperfect Market Models

The objective of this course is to present the study of some stochastic control problems that model market imperfections.

The first chapter deals with the study of investment problems under portfolio constraints. A duality approach is then used to solve this problem.

The second chapter focuses on sequential control problems (switching and impulse control). This class of models is important since it allows us to represent the phenomena related to liquidity problems on the markets. An example of an explicit solution is given in dimension 1 for optimal switching.

Finally, the last chapter presents the singular stochastic control problem. An application is then given for the optimal investment problem in the presence of transaction costs and for the optimal dividend distribution problem.

References

• B. Bouchard et J.-F. Chassagneux «Valorisation de produits dérivées» Economica, 2014

• M. Broadie, J. Cvitanic and H. M. Soner Optimal replication of contingent claims under portfolio constraints, Review of Financial Studies, 11, 59-79 (1998)

• M. Jeanblanc-Picqué et A. N. Shiryaev «Optimization of the flow of dividends», Russian Mathematical Surveys, 50 (2), 257-277, 1995

• W. H. Fleming et H. M. Soner «Controlled Markov Processes and Viscosity Solutions», Second Edition, Stochastic modeling and applied probability, 25, Springer, 2006

• H. Pham «Continuous-time Stochastic Control and Optimization with Financial Applications» Stochastic modeling and applied probability, 61, Springer, 2009

• S. E. Shreve & H. M. Soner «Optimal Investment and Consumption with Transaction Costs», The Annals of Applied Probability, 4(3), 609-692 (1994)