The investigator studies problems in stochastic control and financial mathematics. The first and most important topic is a new look at the dynamic programming method in continuous-time stochastic control using a novel version of Perron's method. Taking the supremum of stochastic sub-solutions and infimum of stochastic super-solutions, the new method provides two viscosity solutions squeezing between them the value function. Uniqueness of the viscosity solution (in case it holds) then easily shows that the value function is the unique solution of the dynamic programming equation. The dynamic programming principle is obtained as a conclusion using this approach, without any a priori analysis of the value function. This amounts to verification without smoothness of the existence of a viscosity solution (similar to the verification argument in the classic case). The second topic of the project resides in understanding the incentives of high-watermark fees on the fund manager, by modeling his/her strategic behavior. The investigator and his colleagues study the optimal choice of the fund manager among available assets (that leads to the fund share price), such that the rational behavior of the investor (utility maximization on her side) yields maximal fees paid to the manager. The third topic is a first step into understanding information percolation in the context of mean-field games of optimal stopping.
Any decision under uncertainty can be modeled as a stochastic control/optimization problem. This applies not only to finance and economics but to engineering and life sciences. The current project mainly consists in a new technical approach to a very general class of stochastic control problems. The new approach provides a deeper understanding of the optimization problems, and it also extends the scope of applications. In addition, the project models and studies the strategic behavior of fund managers, as well as the percolation of information among populations that interact randomly.
