Fractional regularity and stochastic analysis of discretizations. Adaptive algorithm for credit risk simulation.

Summary This thesis concerns three topics in numerical probability and financial mathematics. First, we study the L2 regularity modulus in time of the Z-component of a Markovian RDS with lipschitzian coefficients, but whose terminal function g is irregular. This module is related to the approximation error by Euler scheme. We show, in an optimal way, that the order of convergence is explicitly related to the fractional regularity of g. Next, we propose a sequential Monte Carlo method for efficient pricing of a CDO tranche, based on sequential control variables, in a setting with random recovery rates and i. I. D. Finally, we analyze the hedging error associated with the Delta-Gamma strategy. The fractional regularity of the payoff function plays a crucial role in the choice of rollover dates, in order to achieve optimal convergence speeds.