Mathematical theory of financial markets: equilibria in incomplete reinsurance markets.

Summary This thesis is divided into two parts. The first part is devoted to the study of reinsurance markets in the presence of short selling. We take up the dynamic formalism introduced by Aase(1992), which uses marked point processes to represent the insured risk. We study the concept of competitive equilibrium in this particular type of exchange economy. We introduce the impossibility for companies to reinsure more contracts than they own. This assumption radically changes the nature of the problem, bringing a form of dynamic incompleteness. In the first chapter, we give necessary equilibrium conditions on the reinsurance premium. The second chapter considers the opposite question and gives sufficient conditions for the existence of a competitive equilibrium. The third chapter focuses on the implementation of competitive equilibria through a strategic market game. The second part focuses on the behavior of an investor wishing to hedge a contiguous act, when he does not know perfectly the underlying probabilistic model. We adopt a minimax approach, introduced by Karatzas and Cvitanic (1998). The agent does not know the risk-neutral probabilities. He then assumes that the "Market" is playing against him, by choosing a probability to maximize the hedging error. We thus see a zero-sum game between a fictitious player and the investor. When it admits a value, this value provides an upper bound to the hedging error. We show the existence of this value and analyze its properties. Finally, we consider an unknown parameter (volatility, trend) for which the agent adopts a Bayesian behavior. The two forms of attitudes towards uncertainty (minimax and Bayesian approach) are made to coexist by studying a zero-sum game with asymmetric information, for which we prove the existence of the value.