The goal of this course is the study of continuous time stochastic processes. Stochastic processes is one of the major fields in probability theory and has applications in all the branches of science.
This course will be very helpful for students who wish to take the course SLE in the second semester given by Gady Kozma and Ofer Zeitouni.
The course will have four main parts:
1. Stochastic processes: Definitions and examples. Martingales and Markov processes.
2. Brownian motion: The most basic continuous time stochastic process is Brownian motion.
We will start by building the process and proving basic properties including conformal invariance in two dimensions.
3. It?o calculus: We will construct stochastic integrals and in particular It?o?s integral. We will prove some of the basic properties of it. We will prove It?o?s formula, It?o?s isometry and the
Martingale representation theorem.
4. Applications: There are many applications for stochastic analysis and we will survey only a few of the following (Students requests are welcome) :
(a) Stochastic differential equations.
(b) Solutions of boundary problems: Dirichlet, Poisson, and Neumann problem.
(c) Homogeneous diffusion processes: Definitions, basic properties and Feynman-Kac formula.
(d) Optimal stopping problems.
(e) Brownian motion problems: Local times, winding number and more.
(f) Mathematical finance: Black-Scholes model and more.