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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2006-2007 MATHEMATICS COLLOQUIUM SERIES 02/05/07 Colloquium Dr. Jason Swanson University of Wisconsin, Madison Stochastic Integration with Respect to a Quartic Variation Process Abstract:  Brownian motion (BM) is used to model a wide array of stochasticphenomena in a variety of scientific disciplines. Typically, this is done by using BM as a driving term in a stochastic differential equation (SDE). We areable to define and study these SDEsusing Ito's stochastic calculus. Similarly, stochastic partial differential equations (SPDEs) are often used to model stochastic phenomena. In this talk, we consider a very simple example of a stochastic heat equation. The solution to this SPDE, when regarded as a process indexed by time, has a nontrivial 4-variation. It follows that we cannot use the traditional methodsof the Ito calculus to define an SDE driven by this process. In this talk, I will describe work in progress toward constructing a stochastic integral with respect to this process and a corresponding Ito-like change-of-variables formula. The integral being constructed is a limit of discrete Riemann sums. It turns out that the process we are considering has a very close relationship to a certain "flavor" of fractional Brownian motion(FBM). The quest for a calculus for FBM has led researchers in several different directions and there is a large body of literature on the topic. I will discuss some of the connections between our integral and an analogous approach for FBM. Part of this project is joint work with Chris Burdzy.