Stochastic Calculus and Applications (L24)

• Review of Brownian motion. Isonormal process. Wiener’s existence theorem. Sample path properties. • Continuous stochastic calculus. Martingales, local martingales and semi-martingales. Quadratic variation and co-variation. Ito’s isometry and definition of stochastic integral. Kunita–Watanabe’s theorem. Ito’s formula. • Applications to Brownian motion. Levy’s characterization of Brownian motion. Dubins– Schwartz theorem. Girsanov’s theorem. Transience and recurrence. Martingale representation theorems. • Stochastic differential equations. Strong and weak solutions. Notions of existence and uniqueness. Yamada–Watanabe theorem. Strong Markov property. Kolmogorov, Fokker– Planck and Feynmann–Kac partial differential equations. The one-dimensional case. Stochastic partial differential equations.