Processus stochastiques et équations aux dérivées partielles

Abstract

The aim of this work is to show the relation between the partial differential equations of the second order and the stochastic processes of diffusion, and present some results obtained recently on the partial derivative equations by probabilistic methods. These results provide a probabilistic method that we allow to avoid the complication of numerical methods and written the solution as expectation of functional of diffusion process. This work is presented in five chapters: The chapter I, present the basic mathematics tools, the Brownian motion and the stochastic process solution of stochastic differential equation (SDE) well‐known with noun of diffusion process i.e. that their future is not depending of any other state excepting the present state, is key notion of this study. We introduce a new character of integral, is the stochastic integral says Itô integral that allow to give a sense to the differential of Brownian motion, the important notion upon rest the SDE theories. In the chapter II, we give the generality of partial differential equations (PDE) of second order and explain the method of finite difference method this method is used in case where the resolution by the analytic method is impossible. In the chapter III, we exhibit the profound relation existed between the notion of partial differential equations and stochastic differential equations through the certain theory (Feynman‐Kac), the generalization of this theory given a probabilistic interpretation of PDEs. The chapter IV is an application which we help to comprehend the notions of the president chapter, we start by simulating the trajectory of Brownian motion, and next, we simulate the diffusion process and resolve a PDE by the probabilistic method. The chapter V, it is an application in finance, where we applied the Black and Scholes formula by different methods .