ABSTRACT
In 1944, K. Ito first defined the stochastic integrals of adapted measurable processes with respect to a Brownian motion. The key character of this kind of stochastic integrals is that the processes produced by integration are martingales. This chapter introduces the definition and fundamental properties of stochastic integrals. It presents the very useful Lenglart’s inequality, and by means of it study the continuity of stochastic integrals with respect to integrand processes. The chapter deals the change of variables formula and Doléans-Dade exponential formula for semimartingales and introduces local times of semimartingales and a generalization of Ito formula. It focuses on stochastic differential equations, by using Metivier-Pellaumail’s approach. The chapter defines (indefinite) stochastic integrals of predictable processes w.r.t. local martingales such that the resulted integrals are still local martingales. At first, for elementary predictable processes we can define stochastic integrals in a natural manner.
