Introduction to continuous time stochastic processes. Brownian motions: explicit constructions, properties, quadratic variation, the Cameron-Martin-Girsanov formula, multidimensional Brownian motions. Stochastic Integration: definition, Ito’s formula, martingale representation, time change, Girsanov’s Theorem, local time and Tanaka’s formula. Stochastic differential equations and diffusion processes: strong and weak solutions, diffusions, the Feynman-Kac formula, backward stochastic differential equations. Levy processes: definition, the Levy-Khinchin representation, the Levy-Ito decomposition, the Esscher transform.