The course presents the basic material of a third level of stochastic calculus. It will include the following chapters.

• -  Motivations.

• -  Stochastic differential equations without Lipschitz coefficients: strong existence, pathwise uniqueness, existence and uniqueness in law. Engelbert-Schmidt criterion. Non-explosion conditions

• -  Bessel processes and Cox-Ingersoll-Ross model.

• -  Backward stochastic differential equations and connections with semilinear PDEs.