Basic concepts of stochastic processes, Brownian motion, Gaussian white noise. Conditional expectations and their properties, martingale processes. Stochastic integrals, motivations for the Ito stochastic integral. Ito stochastic integral for simple processes and the general case. Ito Lemma and its different versions. Introduction to stochastic differential equations (s.d.e.) . Solving the Ito s.d.e. by the Ito Lemma and the Stratonovich integration. Homogeneous equations with multiplicative noise. The general s.d.e. with additive noise. A short excursion into finance. Option pricing problem, the Black and Scholes formula.