Skip process and default risk.

Summary This thesis is composed of two parts: in the first part, we study a complete market whose risky asset is a discontinuous process. The second part is devoted to a default risk model. We emphasize the difference between default and non-default market information. Chapters four and five deal respectively with the case where the information is coarse and the case where the default time is a stopping time for the filtration generated by the information available in the default-free market. In the next chapter, we study the conservation property of martingales (assumption (H)). In this framework, we establish a predictable representation theorem and make the link between hypothesis (H) and the absence of arbitrage. The next two chapters generalize these results to the presence of several default moments and to the case where hypothesis (H) is not verified. The ninth chapter studies the incompleteness generated by the default. In particular, we characterize the set of equivalent measure martingales. We determine the range of prices for some contingent assets. Using the predictability theorems, we show that the market can be completed by a zero-coupon with default and we explain the hedging of assets. The last chapter is first devoted to the problem of optimizing the expected wealth utility in the presence of a default. We show that the use of a utility function allows the agent to set a unique martingale equivalent measure. Then, we solve a maximization problem with a random horizon.