We will introduce continuous time martingales, stochastic integration, and basic tools in stochastic analysis including Ito’s formula, various inequalities for local martingales and for stochastic integrals. We will also introduce stochastic differential equations and study their basic properties. Time permitting, we will also discuss completeness, strong completeness of SDEs and differentiation of probabilistic semi-groups.
Revuz and Yor, Continuous Martingales and Brownian Motion
Ikeda and Watanabe: Stochastic Differential Equations and Diffusion Processes