Section: New Results

Backward stochastic (partial) differential equations with jumps, optimal stopping and stochastic control with nonlinear expectation

Participants : Agnès Bialobroda Sulem, Roxana Dumitrescu, Marie-Claire Quenez [(Univ Paris 7)] , Bernt Øksendal, Arnaud Lionnet.

Nonlinear pricing in imperfect financial markets with default.

We pursue the development of the theory of stochastic control and optimal stopping with nonlinear expectation induced by a nonlinear BSDE with (default) jump, and the application to nonlinear pricing in financial markets with default. To that purpose we have studied nonlinear BSDE with default and proved several properties for these equations. We have also addressed the case with ambiguity on the model, in particular ambiguity on the default probability. In this context, we study robust superhedging strategies for the seller of a game optimal stopping problem by proving some duality results, and characterize the robust seller's price of a game option as the value function of a mixed generalized Dynkin game.

Stochastic control of mean-field SPDEs with jumps

We study stochastic maximum principles, both necessary and sufficient, for SPDE with jumps with a general mean-field operator.

Numerical methods for Forward-Backward SDEs

The majority of the results on the numerical methods for FBSDEs relies on the global Lipschitz assumption, which is not satisfied for a number of important cases such as the Fisher-KPP or the FitzHugh-Nagumo equations. In a previous work, A. Lionnet with Gonzalo Dos Reis and Lukasz Szpruch showed that for BSDEs with monotone drivers having polynomial growth in the primary variable y, only the (sufficiently) implicit schemes converge. But these require an additional computational effort compared to explicit schemes. They have thus developed a general framework that allows the analysis, in a systematic fashion, of the integrability properties, convergence and qualitative properties (e.g. comparison theorem) for whole families of modified explicit schemes. These modified schemes are characterized by the replacement of the driver by a driver that depends on the time-grid, and converge to the original driver as the size of the time-steps goes to 0. The framework yields the convergence of some modified explicit scheme with the same rate as implicit schemes and with the computational cost of the standard explicit scheme [55].