Stochastic differential equations

1. Itô's stochastic integral as a random variable and as a process, Itô's formula, Itô representation and the martingale representation theorem. Examples and applications.
2. Strong solution of stochastic differential equations under the Lipschitz condition: existence and uniqueness. Examples and applications.
3. Diffusion processes: infinitesimal generator, Dynkin's formula, Kolmogorov's backward and forward equations. Examples and applications.
4. Diffusion processes and related elliptic and parabolic partial differential equations (Laplace, Poisson, Helmholtz equations; heat equation, Feynman-Kac formula.) Examples and applications.
5. Change of measure and Girsanov's theorem, diffusion processes as martingales under an equivalent measure. Examples and applications.