Special Colloquium
Jason Swanson
University of Wisconsin-Madison
Stochastic integration with respect to a quartic variation process
Abstract: Fractional Brownian motion (fBm) has presented itself as an attractive
alternative to Brownian motion (BM) in many applied areas, including the
analysis of computer network traffic and mathematical finance. This is
because, unlike BM, fBm is capable of modeling long-range dependence in
time. The classical stochastic calculus of Itô, which allows us to define
and study stochastic differential equations driven by BM, does not extend to
fBm. The quest for a calculus for fBm has led researchers in several
different directions.
In this talk, we consider the solution, u(t,x), to a certain stochastic heat
equation. For fixed x, the process F(t)=u(t,x) shares many properties with
fBm and we will discuss a new approach to constructing a calculus for F. The
difficulties in this construction stem from the roughness of the paths of F.
Specifically, F has a nontrivial 4-variation. Our construction develops an
integral with respect to F which is the limit, in distribution, of a
sequence of discrete Riemann sums.
Thursday December 14, 2006 at 3:00 PM in SEO 636