Friday, March 14, 2014 at 3:30pm
Frank H. T. Rhodes Hall, 655
CAM Colloquium: Pierre Patie (Cornell) - Boundary crossing problems for the Brownian motion: A calculator viewpoint
The problem of finding the distribution of the first crossing time of a Brownian motion to a moving boundary is an old and classic problem in the theory of stochastic processes. Fine distributional properties of this stopping time are substantial in statistical analysis, describing stochastic neuronal activity, epidemiology, pricing barrier and American options in finance and many other areas. Despite this long history and wide range of applications, few closed-form solutions are available. In this talk, we start by reviewing the different approaches which have proved to be efficient for solving this problem for specific curves. We proceed by describing a new methodology which allows to connect, by means of a simple analytical expression, the law of the first crossing time of a Brownian motion over a curve to a two-parameters family of curves. We shall indicate several directions to prove this fact highlighting the connection between this stochastic problem, functional analysis and Lie group techniques elaborated to study the heat equation. We end this talk by discussing some open problems.