KAZI TANI Mohamed Nabil

This doctoral dissertation focuses on the modeling of preventive effort and its relationship with market insurance. Each chapter attempts to capture different aspects of this problem, from the study of a criterion consistent with actuarial practices to the study of the supply side of insurance, including risk perception biases and an approach to prevention in dynamic time. Chapter 1 models the relationship between an insurer and an insured as a Stackelberg game. In this game, the insurer plays first by offering an insurance contract in the form of a loading factor. The insured then plays by choosing the optimal coverage rate and prevention effort. Both the insured and the insurer aim to minimize their respective risk measures, which are both consistent. The respective effects of self-insurance and self-protection on risk minimization will be studied. In each case, it will be shown that optimal choices for the insured exist and the optimal contract for the insurer will be characterized. Moreover, it will be shown that if the agent's risk measure decreases faster than his loss expectation, then the optimal effort is increasing with the loading factor with a potential discontinuity when the optimal coverage goes from full to zero. However, in the opposite case the optimal effort can be increasing or decreasing with the loading factor. Chapter 2 studies the relationship between self-insurance and market insurance also in the form of an optimization problem for one agent. Similar to Chapter 1, this agent must determine the coverage rate and the prevention effort that will optimally reduce its risk measure. The considered risk measure is called distortional and is defined from a non concave distortion function. This allows for potential individual cognitive biases in risk perception. The characterization of the optimal solution for the agent makes it possible to draw a new conclusion about the relationship between self-insurance and market insurance. Self-insurance is no longer just a substitute for market insurance, but can also be complementary to it, depending on the sensitivity of the prevention effort to the price of insurance. Chapter 3 focuses on self-protection by proposing a dynamic expected utility maximization problem. This takes the form of a stochastic control problem in which the agent chooses his insurance coverage and his prevention effort which is dynamic. The problem can be separated into two subproblems, the first one is an optimization in effort and the second one in insurance coverage. Since the individual wants to obtain the largest possible final wealth, he seeks to maximize the exponential utility expectation of this wealth. The agent's wealth can be seen as the solution of a backward-looking stochastic differential equation with a jump, this equation admits a unique solution and is moreover explicit. In particular, we obtain that the optimal self-protection effort is constant. The initial distribution of the loss process, when there is no effort, is given by a compound Poisson process which is in particular a Lévy process. Obtaining a constant optimal effort means that the Lévy property of the processes is preserved by maximizing an exponential utility expectation. The analysis of the problem in insurance coverage gives a sufficient condition to obtain the existence of an optimal level of coverage. The individual can then subscribe to an insurance policy by providing a preventive effort that will maximize his satisfaction or choose not to subscribe to the policy but by taking part in self-protection actions.

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.