ST 651

Stochastic processes based on Brownian motion are the central object of the course. Applications of stochastic differential equations (SDEs) in Science and Finance are discussed. The topics include: Brownian motion (BM) and its derivatives such as Brownian bridge and Geometric Brownian motion (GBM), Gaussian processes, simulation of stochastic processes, the Riemann-Stieltjes integrals and stochastic integrals with Brownian motion, Itô processes and Itô’s formula, the Feynman-Kac theorem, multivariate BM and GBM, bivariate Itô’s formula, solution of SDEs analytically using the Itô’s formula and numerically using the Euler and Milstein schemes, standard and compound Poisson processes, jump-diffusion processes, and simulation of processes with jumps. The optional topics include the first-passage time distribution for Brownian motion. <br>