School Seminars and Colloquia

Boundary Crossing Probabilities for Diffusion Processes and Related Problems

Stochastic Processes Seminar

by Andrew Downes

Institution: The University of Melbourne
Date: Fri 22nd August 2008
Time: 1:15 PM
Location: Russell Love Theatre, Richard Berry Bldg, The Uni of Melbourne

Abstract: Calculating the probability that a diffusion process will stay under a
> given curvilinear boundary during a given time interval is of great
> importance for applications including financial mathematics and sequential
> analysis. Since closed-form solutions are in general unknown finding
> approximate solutions is of substantial interest. One possible approach is
> to approximate the given boundaries with `close' ones, which have a more
> tractable boundary crossing probability. Intuitively the resulting
> crossing probability should be a good approximation for the original
> problem, but it is important to quantify the accuracy of this approach.
> The main results of this thesis include upper bounds for the approximation
> rate.
> Under mild regularity conditions we show that the difference between the
> probabilities does not exceed an explicit multiple of the uniform distance
> between the original and approximating boundaries.
> We also establish the existence of the first crossing time densities and
> provide new sharp bounds for them. A similar approach enables us to obtain
> further results, including new sharp bounds for transition densities of
> diffusion process, and an exact relationship between the asymptotic form
> of the crossing probability for the pinned process and the first passage
> density.
> The work is also extended to jump-diffusion processes.

For More Information: Contact: Aihua Xia or Daniel Dufresne