Boundary Crossing Probabilities for Diffusion Processes and Related Problems
by Andrew Downes
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
> 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
> The work is also extended to jump-diffusion processes.
For More Information: Contact: Aihua Xia A.Xia@ms.unimelb.edu.au or Daniel Dufresne firstname.lastname@example.org