Section: New Results

Optimal stopping

Our research on optimal stopping problems covers the analysis of free boundaries in optimal stopping problems for multidimensional stochastic processes with jumps (Thesis of A. Bouselmi, supervised by D. Lamberton). Numerical issues are also be investigated (Monte Carlo methods, quantization methods, methods based on Malliavin calculus). Even in diffusion models, a realistic dividend modeling introduces jumps in the dynamics : at the dividend dates the spot value of the stock undergoes a jump equal to minus the dividend amount. We plan to take into account this feature in optimal stopping problems (Thesis of M. Jeunesse, supervised by B. Jourdain).

The pricing of American options with irregular payoff such as, for instance, binary options, leads to challenging mathematical problems. Some theoretical properties of optimal stopping problems with irregular payoffs have already been obtained. We now plan to focus on the Markovian case by using viscosity solutions and numerical analysis techniques.

In [45] , we study optimal stopping problems for (non necessarily) convex dynamic risk measures induced by BSDEs with jumps and establish their connections with Reflected BSDEs with jumps. Such problems are related to optimal stopping for non linear expectations, which has been recently studied by [58] in the convex case only. We also address the case of model ambiguity and its relation with mixed control/optimal stopping problems.