Filtration enlargement with applications to finance.

Summary This thesis consists of four independent parts. The main thread of this one is the filtration magnification. In the first part, we present classical results of filtration magnification in discrete time. We study some examples in the context of initial filtration magnification. In the framework of progressive magnification we give conditions to obtain the immersion property of martingales. We also give various characterizations of pseudo stopping times and we state properties for honest times.In the second part, we are interested in the pricing of variable annuity products in the context of life insurance. For this we consider two models, in both of which we consider the market to be incomplete and adopt the indifference pricing approach. In the first model we assume that the insured makes random withdrawals and we compute the indifference premium by standard methods in stochastic control. We solve stochastic backward differential equations (SDEs) with a jump. We provide a verification theorem and we give the optimal strategies associated with our control problems. From these, we derive a computational method to obtain the indifference premium. In the second model we propose the same approach as in the first model but we assume that the insured makes withdrawals that correspond to the worst case for the insurer. In the third part, we study the relationship between the EDSR solutions in two different filterings. We then study the relationship between these two solutions. We apply these results to obtain the indifference price in the two filtrations, i.e. the price at which an agent would have the same level of expected utility using additional information.In the fourth part, we consider advanced stochastic backward differential equations (EDSRAs) with one jump. We study the existence and uniqueness of a solution to these EDSRAs. For this purpose we use the decomposition of jump processes related to the progressive coarsening of filtration to bring us back to the study of Brownian EDSRAs before and after the jump time.