Could anyone recommend someting on the first exit time of a Brownian bridge?

braveyellowj · June 20th, 2007, 10:52 am

I'm studying a problem on the probability of a stochastic process X(t), such that it never exits a finite interval [a, b] during time [0,T], and X(0)=x1, X(T)=x2 are fixed constants.That isP {tao_a ^ tao_b > T | X(0)=x1, X(T)=x2 }X(t)'s density is known.I have no clues yet. Could anyone recommend some related materials? For the simplest case X(t) isa brownian bridge.Thank you so much. I would greatly appreciate it.

LordR · June 20th, 2007, 11:08 am

If X is a Brownian motion, I'd suggest you to look at papers dealing with double barrier options.

braveyellowj · June 20th, 2007, 11:11 am

Thank you very much for your reply.I can explicitly calculate the Laplace form of tao_a ^ tao_b (the minimum of the two).The difficulty for me is how to include X(T) = x2?

Leonidas · June 21st, 2007, 6:16 am

Hi,most of the papers relate to the book "Brownian Motion and Stochastic Calculus" by Shreve and Karatzas, where the most important things are available. You will also find the answer in the Paper "Exact Exotics" , published by Andersen Brotherton-Ratcliffe in Risk, or the paper "Going to Extremes: Correcting Simulations Bias in Exotic Option Valuation" by Beaglehole, Dybvig and Guofu Zhou, which can be found on Zhou's homepage. Also, check out this great Script by S. Crepey.

braveyellowj · June 21st, 2007, 4:34 pm

Thank you so much. They are very helpful.