Euler schemes for killed diffusion. Application to barrier options.

Summary This thesis consists of two chapters, devoted to the approximation by euler diagrams of the expectation of a certain functional of the trajectory of a multidimensional diffusion process between time 0 and t. We are interested in the law in t of the diffusion killed at its exit from an open d : the functional in question is equal to a function f of the value of the process in t if the process remains in d between times 0 and t, and is 0 if the process has exited. The motivation for this work comes from financial mathematics, where the evaluation of the price of barrier options comes down to the calculation of this type of expectation. To obtain an approximate value of this expectation, we discretize the diffusion with an approximation scheme and evaluate the expectation of the functional for the scheme by a monte-carlo method. In the first chapter, we consider the euler scheme in continuous time, obtained from a regular subdivision of the time interval, and we analyze the approximation error as a function of the discretization step. The use of the malliavin calculus allows to reach the case where the function f is only measurable. The simulation by monte-carlo method is easy in the one-dimensional case, but becomes more delicate in higher dimension. In the second chapter, we consider the euler scheme in discrete time. In this case, the simulation is easy independently of the dimension, but the approximation error is more important than in the continuous case. The analysis of the error leads us in particular to explain a semimartingale decomposition of the orthogonal projection on the closure of d of a continuous semimartingale.