DMS Graduate Student Seminar

Speaker: Nick Eisenberg 

Title: An introduction to stochastic calculus with applications to finance and SPDEs.

Abstract: Stochastic calculus is a relatively new field that was pioneered by a Japanese mathematician named Kiyosi Ito. In 1944, Ito published a paper titled "Stochastic Integral," which inspired many other mathematicians to research and further expand this area of math. Today, stochastic calculus, and in particular stochastic partial differential equations, is a very vast subject with many open problems. In addition, the field has many applications in disciplines that range from theoretical physics, neuroscience, and biology all the way to finance. 

At the beginning of the talk, I will introduce some fundamental concepts of stochastic calculus. We will first define what a stochastic process is and then give an example through a random walk. We will then use this idea to heuristically construct a Brownian motion which will then be used to construct the Ito stochastic integral. We will then use these ideas to give an application to finance by introducing the concept of a geometric Brownian motion and its relation to the Black Scholes European option pricing model. Lastly, we will give an application to physics by discussing some theoretical results of the stochastic heat equation.