Processus de diffusion et équations différentielles stochastiques -aspects theorique et numérique- applications

Abstract

The differential calculation gives a setting in the notion of ordinary differential equation, that acts as model for variable phenomena in the time. When we wanted to add to these equations random disturptions, we have been embarrassed by the non differentiability of the Brownian movement. The definition of a problem differential stochastic passes by the integral of Itô. The formula of Itô is to the basis of the differential calculation techniques on the stochastic processes that we regroup under the stochastic calculation name. The objective of this work is to present the theory of Itô of the stochastic equations under differential or complete form, and to show the ties that can exist between the stochastic differential equations and the equations to the partial derivatives. The instruments used to reach this objective are essentially the stochastic calculation and the elementary techniques of the equations differential ordinary determinists. We starts with constructing an integral in relation to the Brownian movement, for in continuation to define the notion of stochastic differential equation. The used stochastic processes are the processes possessing the property of Markov. We indicate the probabilistic representations of the solutions of the numerous problems. These representations provide an in instrument of analysis or calculation of approximations of the solutions of these equations