Stochastic Analysis

Lecture:

Stochastic analysis is a fundamental branch of probability theory which is linked to diverse areas of mathematics (e.g. partial differential equations, mathematical physics, geometry) and finds applications in finance, chemistry and biology. The target audience for the course “Stochastic Analysis”  will be students in mathematics who are already familiar with fundamental concepts of probability theory.

As for the content, we will start by laying out different constructions of Brownian motion and discussing some of its fundamental probabilistic, analytic, and statistical properties, e.g., the Markov property. Then, we will be able to define stochastic integrals in the Itô sense, with regard to general continuous local martingales, such as the Brownian motion. We will also discuss the properties of such integrals, such as the Itô formula. We will then provide three concrete applications of this theory, namely  we will

a) construct "local times" of Brownian motion,

b) study stochastic (ordinary) differential equations, i.e., evolution equations that are driven by some random noise and

c)  investigate martingale problems and general continuous time Markov processes.