# Analysis and numerical solution of an inverse first passage problem from risk management

Abstract (Summary)

We study the following "inverse first passage time" problem. Given a diffusion process Xt and a
probability distribution q(t) on t &ge 0, does there exist a boundary b(t) such that q(t)=P[&tau &ge t], where &tau is the first hitting time of Xt to the time dependent level b(t). We formulate the inverse first passage time problem
into a free boundary problem for a parabolic partial differential operator and prove there exists a unique viscosity solution to the associated Partial Differential Equation by using the classical penalization technique. In order to compute the free boundary with a given default probability distribution, we investigate the small time behavior of the boundary b(t), presenting both upper and lower bounds first. Then we derive some integral equations characterizing the boundary. Finally we apply Newton-iteration on one of them to compute the boundary. Also we compare our numerical scheme with some other existing ones.
Bibliographical Information:

Advisor:Xinfu Chen; David Saunders; John Chadam; Soiliou Daw Namoro; Ivan Yotov

School:University of Pittsburgh

School Location:USA - Pennsylvania

Source Type:Master's Thesis

Keywords:mathematics

ISBN:

Date of Publication:07/07/2006