Interprétation probabiliste des EDP et processus de diffusion

Abstract

The objective of this work is to show the bond between the partial derivative equations of the second order and the stochastic processes of diffusion like having some results obtained recently on the partial derivative equations by probabilistic methods. Many solutions of partial derivative equations of the second order can be written like the hope of a functional calculus of a process of diffusion. The probabilistic formulas, often known as formulas of Feynman-Kac, constitute a tool which makes it possible to show many results on the partial derivative equation corresponding by probabilistic methods. The Brownian movement is, amongst other things, the major tool of the model of Black and Scholes and is used to build the majority of the models of credits in finance. The systems appearing in the applications are often subjected to disturbances which one can regard as random. One gives initially the probabilistic interpretation of the equation of heat thanks to the Brownian movement. The relationship between the Brownian movement and the equation of heat is illustrated between certain systems of partial derivative equations, the physical problems from which they result and the stochastic differential equations. One shows then that the results obtained for the Laplacian can be generalized with the operators of the second order with variable coefficients, the Brownian movement, being, in this case, replaced by a process of diffusion. The practical point of view of modelling is studied in particular through the algorithms of numerical resolution of Euler and Milstein by the means of some problems. The stochastic differential equations find an application significant in finance. It is the object of the last part of this work in which one is interested particularly in the evaluation of options in the model of Black and Scholes