Section: New Results

Risk measures, BSDEs with jumps and nonlinear expectations

Participants : Agnès Sulem, Marie-Claire Quenez, Z. Chen [Shandong University].

In the Brownian case, links between dynamic risk measures and Backward Stochastic Differential Equations (BSDEs) have been established. A. Sulem and M.-C. Quenez are exploring these links in the case of stochastic processes with jumps. They have extended some comparison theorems for BSDEs with jumps, and provided a representation theorem of convex dynamic risk measures induced by BSDEs with jumps. They 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. They also study the case of model ambiguity and its relation with mixed control/optimal stopping problems.

There are two classes of nonlinear expectations, one is the Choquet expectation given by Choquet (1955), the other is the Peng's $g$-expectation given by Peng (1997) via backward differential equations (BSDE). Recently, Peng raised the following question: can a $g$-expectation be represented by a Choquet expectation? In [26] , A. Sulem and Z. Chen provide a necessary and sufficient condition on $g$-expectations under which Peng's $g$-expectation can be represented by a Choquet expectation for some random variables (Markov processes). It is well known that Choquet expectation and $g$-expectation (also BSDE) have been used extensively in the pricing of options in finance and insurance. Our result also addresses the following open question: given a BSDE ($g$-expectation), is there a Choquet expectation operator such that both BSDE pricing and Choquet pricing coincide for all European options? Furthermore, the famous Feynman-Kac formula shows that the solutions of a class of (linear) partial differential equations (PDE) can be represented by (linear) mathematical expectations. As an application of our result, we obtain a necessary and sufficient condition under which the solutions of a class of nonlinear PDE can be represented by nonlinear Choquet expectations [26] .