Section: New Results

Dynamic risk measures and BSDEs with jumps

The standard approach of mathematical quantification of financial risk in terms of Value at Risk has serious deficiencies. This has motivated a systematic analysis of risk measures which satisfy some minimal requirements of coherence and consistency. The theory of risk measures has been first developed in [54] in the coherent case and then extended in various directions (convex, dynamic, law-invariant) (see e.g. [70] , [68] , [93] , [69] ). We are extending this theory, in particular in the case of markets with possible random jumps and model ambiguity, and investigate various types of optimization problems involving risk measures.

Mathematical techniques for the treatment of such problems are based on non linear expectations, backward stochastic differential equations (BSDEs), stochastic control, stochastic differential games.

In the Brownian case, links between dynamic risk measures and Backward Stochastic Differential Equations (BSDEs) have been established (see, among others, [57] ). A. Sulem and M.-C. Quenez are exploring these links in the case of stochastic processes with jumps. To this purpose, we have recently extended some comparison theorems for BSDEs with jumps given in [90] , and provided a representation theorem of convex dynamic risk measures induced by BSDEs with jumps (see [44] ). Optimization of dynamic risk measures leads to stochastic differential games or to optimal control problems for coupled systems of forward-backward stochastic differential equations (FBSDEs). They can be studied by stochastic maximum principles [100] or by transforming them into controlled Backward Stochastic Partial Differential Equations (BSPDEs). We address these questions in collaboration with B. Øksendal (Oslo university) and T. Zhang (Manchester University).

The numerical study of (F)BSDEs with jumps is especially demanding in high dimensions and collaboration has started on these issues with J. Lelong (ENSIMAG) and C. Labart (Université de Savoie).