Reflective processes in finance and numerical probability: regularities and approximation of reflective SDEs and American options in the presence of transaction costs.

Summary This thesis is composed of two independent parts that focus on the application of probability to the field of finance. The first part studies the regularity of the solutions of certain types of backward-looking stochastic differential equations (SRDEs) and reflexive differential equations, as well as numerical approximation schemes of these solutions. In finance, the main application is the pricing and hedging of American and gambling options, but our work is not limited to this framework. The proposed systematic method is based on the study of equations that are reflected only on a discrete time grid. In finance, these equations are interpreted as Bermuda options. In a general framework of multidimensional convex domains that can, under certain conditions, evolve randomly, we obtain convergence and regularity results for these discretely reflected equations that we extend to continuously reflected SDEs. The second part deals with a theoretical problem in mathematical finance. We deal with the valuation of American options in the framework of market models with proportional transaction costs, both for discrete and continuous time. We obtain an over-replication theorem for these contingent assets in the very general framework of ladlag option processes.