Introduction to Stochastic Calculus and to the Resolution of PDEs Using Monte Carlo Simulations

Abstract

I give a pedagogical introduction to Brownian motion, stochastic calculus introduced by Ito in the fifties, following the elementary (at least not too technical) approach by Follmer [Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 143–150. Springer, Berlin, 1981]. Based on this, I develop the connection with linear and semi-linear parabolic PDEs. Then, I provide and analyze some Monte Carlo methods to approximate the solution to these PDEs. This course is aimed at master students, Ph.D. students and researchers interesting in the connection of stochastic processes with PDEs and their numerical counterpart. The reader is supposed to be familiar with basic concepts of probability (say first chapters of the book Probability essentials by Jacod and Protter [Probability Essentials, 2nd edn. Springer, Berlin, 2003]), but no a priori knowledge on martingales and stochastic processes is required.