This book is a thorough and self-contained treatise of martingales as a tool in stochastic analysis, stochastic integrals and stochastic differential equations. The book is clearly written and details of proofs are worked out.
Inhaltsverzeichnis
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.
