## Departmental Colloquium

David Nualart

University of Kansas

Stochastic calculus with respect to the fractional Brownian motion

**Abstract:**The fractional Brownian motion is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H$ in $(0,1)$ called the Hurst index. We will first describe some basic properties of the fractional Brownian motion such as long-range dependence and finite p-variation. The applications of the fractional Brownian to model data coming from engineering, finance and other areas, require the construction of a suitable stochastic calculus, similar to the classical Ito calculus. In this talk we review some recent results on the stochastic calculus with respect to the fractional Brownian motion with emphasis on the construction of stochastic integrals using different types of Riemann sums approximations. We will present central limit results for critical values of the Hurst parameter where the approximation diverges, and we will discuss numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$.

Friday October 11, 2013 at 3:00 PM in SEO 636