Fundamentals Stochastic Analysis
Main topics are: Brownian motion (Wiener process), martingales, stochastic (Ito) integration, stochastic differential equations, diffusion processes. These tools are heavily used in financial mathematics, biology, physics, and engineering. Thus if someone wants to enter e.g. the flourishing field of financial mathematics, it is a must to complete such a course.
The goals of the course: Review of some calculus and probability tools. A review of the theory of stochastic processes, including continuous time Markov processes. Introducing the student to the major topics of stochastic calculus, including stochastic integration, stochastic differential equations and diffusion processes. Introducing to some applications, in particular the Black-Scholes model of financial mathematics. The learning outcomes of the course: A good understanding of continuous time stochastic processes, including Wiener process and other diffusion processes (Ito diffusions). Understanding and competence in stochastic 116 integration and stochastic differential equations (SDE’s), strong and weak solutions, and conditions for existence and uniqueness. Practice in solving linear SDE’s, understanding the Ornstein-Uhlenbeck process. Understanding the relationship between weak solutions and the Stroock-Varadhan martingale problem; the notion of generator of a diffusion, and the related backward and forward partial differential equations.
- regular homework, and presentation or final