Stochastic Analysis and Diffusion Processes

Abstract

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.