Contribution to numerical methods applied to finance and differential games.

Summary This thesis presents a combination of partial differential equation and probability techniques for various applied mathematical problems. Regularization methods are used to approach these problems numerically. The first part, differential games, aims at the numerical determination of the (discontinuous) barriers of the evasion pursuit games, via the introduction of auxiliary games in minimum distance. We propose a numerical scheme on a destructured mesh for the solutions of such games. The numerical approach is then validated on an analytically solved game. Finally, we establish a convergence result by probabilistic methods a la Kushner for differential game problems. The second part, financial mathematics, deals with numerical methods by partial differential equations in finance - direct methods, including the study of transparent edge conditions or of an American put on arithmetic mean, and inverse methods, with the study of a problem of calibration of the dynamics of an underlying asset from the prices of derivative products observed on the markets. If the pure calibration problem is underdetermined, it becomes on the other hand well posed by harmonic regularization, in a tikhonov type approach, as it follows from qualitative properties of call prices and bounds on their sensitivities in a black-scholes model with local volatility.