In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

Ce travail étudie des modèles financiers pour les prix d'options, les taux d'intérêts et le risque de crédit, avec des processus stochastiques à mémoire et comportant des discontinuités. Ces modèles sont formulés en termes du mouvement Brownien fractionnaire, du processus de Lévy fractionnaire ou filtré (et doublement stochastique) et de leurs approximations par des semimartingales. Leur calcul stochastique est traité au sens de Malliavin, et des formules d'Itô sont déduites. Nous caractérisons les probabilités risque neutre en termes de ces processus pour des modèles d'évaluation d'options de type de Black-Scholes avec sauts. Nous étudions également des modèles de taux d'intérêts, en particulier les modèles de Vasicek, de Cox-Ingersoll-Ross et de Heath-Jarrow-Morton. Finalement nous étudions la modélisation du risque de crédit / This work investigates financial models for option pricing, interest rates and credit risk with stochastic processes that have memory and discontinuities. These models are formulated in terms of the fractional Brownian motion, the fractional or filtered Lévy process (also doubly stochastic) and their approximations by semimartingales. Their stochastic calculus is treated in the sense of Malliavin and Itô formulas are derived. We characterize the risk-neutral probability measures in terms of these processes for options pricing models of Black-Scholes type with jumps. We also study models of interest rates, in particular the models of Vasicek, Cox-Ingersoll-Ross and Heath-Jarrow-Morton. Finally we study credit risk models