Analysis of Backward SDEs with Jumps and Risk Management Issues.

Summary This thesis deals on the one hand with risk management, measurement and transfer issues and on the other hand with problems of stochastic analysis with model uncertainty. The first chapter is devoted to the analysis of Choquet integrals as monetary risk measures that are not necessarily law invariant. We first establish a new representation result for comonotone risk measures, then a representation result for Choquet integrals by introducing the notion of local distortion. This allows us to give an explicit form to the inf-convolution of two Choquet integrals, with examples illustrating the impact of the absence of the law invariance property. We then consider a pricing problem for a non-proportional reinsurance contract, containing reconstitution clauses. After defining the indifference price relative to both a utility function and a risk measure, we frame it with easily implementable values. We then move to a dynamic time framework. For this purpose, we show, by adopting a fixed point approach, an existence theorem of bounded solutions for a class of stochastic backward differential equations (SDEs in the following) with jumps and with quadratic growth. Under an additional classical assumption in the framework with jumps, or under a convexity assumption of the geKazi-Taninator, we establish a uniqueness result through a comparison principle. We analyze the properties of the corresponding nonlinear expectations. In particular, we obtain a Doob-Meyer decomposition of the nonlinear supermartingales as well as their regularity in time. As a consequence, we easily deduce an inverse comparison principle. We apply these results to the study of dynamic risk measures associated, on a filtration generated by both a Brownian motion and a random integer measure, to their dual representation, as well as to their inf-convolution, with explicit examples. The second part of this thesis deals with the analysis of model uncertainty, in the particular case of second order EDSRs with jumps. We impose that these equations take place in an almost-sure sense, for a whole family of non-dominated probability measures which are solutions of a martingale problem on the Skorohod space. We first extend the definition of second order EDSRs, as defined by Soner, Touzi and Zhang, to the case with jumps. To do so, we prove an aggregation result in the sense of Soner, Touzi and Zhang on the space of càdlàg trajectories. This allows us, among other things, to use a quasi-safe version of the compensator of the canonical process jump measure. We then show an existence and uniqueness result for our class of second order EDSRs. These equations are affected by the uncertainty on both the volatility and the jumps of the process that drives them.