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Analysis of Backward SDEs with Jumps and Risk Management Issues

Abstract : This PhD dissertation deals with issues in management, measure and transfer of risk on the one hand and with problems of stochastic analysis with jumps under model uncertainty on the other hand. The first chapter is dedicated to the analysis of Choquet integrals, as non necessarily law invariant monetary risk measures. We first establish a new representation result of convex comonotone risk measures, then a representation result of Choquet integrals by introducing the notion of local distortion. This allows us then to compute in an explicit manner the inf-convolution of two Choquet integrals, with examples illustrating the impact of the absence of the law invariance property. Then we focus on a non-proportional reinsurance pricing problem, for a contract with reinstatements. After defining the indifference price with respect to both a utility function and a risk measure, we prove that is is contained in some interval whose bounds are easily calculable. Then we pursue our study in a time dynamic setting. We prove the existence of bounded solutions of quadratic backward stochastic differential equations (BSDEs for short) with jumps, using a direct fixed point approach. Under an additional standard assumption, or under a convexity assumption of the generator, we prove a uniqueness result, thanks to a comparison theorem. Then we study the properties of the corresponding non-linear expectations, we obtain in particular a non linear Doob-Meyer decomposition for g-submartingales and their regularity in time. As a consequence of this results, we obtain a converse comparison theorem for our class of BSDEs. We give applications for dynamic risk measures and their dual representation, and compute their inf-convolution, with some explicit examples, when the filtration is generated by both a Brownian motion and an integer valued random measure. The second part of this PhD dissertation is concerned with the analysis of model uncertainty, in the particular case of second order BSDEs with jumps. These equations hold P-almost surely, where P lies in a wide family of probability measures, corresponding to solutions of some martingale problems on the Skorohod space of càdlàg paths. We first extend the definition given by Soner, Touzi and Zhang of second order BSDEs to the case with jumps. For this purpose, we prove an aggregation result, in the sense of Soner, Touzi and Zhang, on the Skorohod space D. This allows us to use a quasi-sure version of the canonical process jump measure compensator. Then we prove a wellposedness result for our class of second order BSDEs. These equations include model uncertainty, affecting both the volatility and the jump measure of the canonical process.
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Submitted on : Tuesday, January 29, 2013 - 11:30:43 AM
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Mohamed Nabil Kazi-Tani. Analysis of Backward SDEs with Jumps and Risk Management Issues. Probability [math.PR]. Ecole Polytechnique X, 2012. English. ⟨pastel-00782154⟩

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