Numerical problems from calculus of variations and mathematical finance.

Summary The present work is oriented in two research directions. It has been realized under the direction of Ivar Ekeland. The first part is placed in the framework of the theory of Hamiltonian systems. We use a variational approach and the principle of Clarke duality, the aim of the study being to develop numerical methods to find the solutions of first and second order equations in order to study their index. Two methods are presented, one giving a large number of solutions but this at the price of the absence of control on the obtained solutions (in particular on their index), the other one linked to the neck theorem, of Ambrosetti and Rabinowitz, ensuring to find solutions of index 1. In this last case we have looked for asymptotic estimates of the solutions found on an example. The second part deals with problems of financial mathematics, concerning the valuation of contingent assets, a field that has developed strongly in recent years. This part was co-directed by Guy Barles. On the one hand, we consider the study of a two-factor Markov model for contingent assets. We present the implementation of a numerical method. On the other hand, we consider an option pricing model depending on the history of the underlying price. The numerical implementation of the model is also presented.