Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, January 25, 2017.

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Wednesday, January 25, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, January 25, 2017

Degenerate diffusions and heat kernel estimates

Jing Wang (J.L. Doob Research Assistant Professor, University of Illinois at Urbana-Champaign)

Abstract: In this talk we will look at degenerate hypoelliptic diffusion processes and the small time behaviors of their transition densities. Diffusion processes play important roles in modeling risky assets in financial mathematics and actuarial science. The small time estimates of their transition densities are particularly useful for pricing options with short maturities. In this talk we will introduce the degenerate diffusion processes that are characterized by their levels of degeneracy. The ones of weaker degeneracy -- also called strong Hörmander's type -- are closely related to sub-Riemannian geometry. An important example is the Brownian motion process on a sub-Riemannian manifold. In general, small time asymptotic estimates are available for a subelliptic heat kernel on the diagonal and out of cut-locus. In special cases such as for Brownian motions on sub-Riemannian model spaces, we can obtain explicit expressions for their transition densities (heat kernels) and hence small time asymptotic estimates, particularly on the cut-loci. In the second part of the talk, we will study the strictly degenerate case-diffusion processes that are of weak Hörmander's type. Namely the hypoellipticity is fulfilled with the help of the drift term. This type of processes are particularly interesting in financial mathematics for pricing Asian options. We obtain large deviation properties for nilpotent diffusion processes of weak Hörmander's type.

5:15 pm in 245 Altgeld Hall,Wednesday, January 25, 2017

IGL Outreach Information Session

  [email]

Abstract: Come learn about activities the Illinois Geometry Lab runs in the local community throughout the year, and how to get involved!