Martingales and Stochastic Analysis

Abstract

Part 1 Stochastic processes: generated theta-algebras stochastic processes stopping times convergence in Lp and uniform integrability. Part 2 Martingales: martingale, submartingale and supermartingale fundamental submartingale inequalities convergence of submartingales uniformly integrable submartingales regularity of sample functions of submartingales increasing processes. Part 3 Stochastic integrals: L2-martingales and quadratic variation processes stochastic integrals with respect to martingales Ft-Brownian motions local martingales and extensions of the stochastic integral Ito's formula Ito's stochastic calculus. Part 4 Stochastic differential equations: the space of continuous functions on R++ definition and function space representation of solutions existence and uniqueness of solutions strong solutions.