lecture27 - ∞ = 1 Let V t = exp-2 μX t which is a...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
FIRST HITTING PROBABILITY WITH TWO BARRIERS XIXI WANG, CITED FROM THE BOOK “APPLIED STOCHASTIC PROCESS” BY YUANLIE LIN, TSINGHUA UNIVERSITY PRESS Abstract. In class we’ve studied the first hitting probability for a Brownian motion with drift. I have found in another book an alternative way to solve this through a martingale approach. 1. problem Assume { X ( t ) = B ( t ) + μt, t 0 } is Brownian Motion with drift rate μ . For a, b > 0, - b < x < a , T a = min { t : t > 0 , X ( t ) = a } , T - b = min { t : t > 0 , X ( t ) = - b } . We have (1.1) P ( T a < T - b < ∞| X (0) = x ) = exp { 2 μb } - exp {- 2 μx } exp { 2 μb } - exp {- 2 μa } 2. proof Assume X (0) = 0. Let T ab = inf { t > 0; X ( t ) = a or X ( t ) = b } and T n = min { T, n } , T = T a ( - b ) . As B ( t ) is martingale, and T n ( n 1) is stopping time of B ( t ), and P ( T n < ) = 1. Therefore (2.1) 0 = E [ B (0)] = E [ B ( T n )] = E [ X ( t n )] - μE [ T n ] . so E [ T n ] 1 μ E [ X ( T n )] 1 μ ( a + b ) < , thus for n 1, E [ T n ] 1 μ ( a + b ) < . And n 1, T n T ( n + 1), from monotone convergence theorem, (2.2) E ( T ) = lim n →∞ E ( T n ) 1 μ ( a + b ) < . So P ( T <
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ∞ ) = 1. Let V ( t ) = exp {-2 μX ( t ) } , which is a martingale. As T a (-b ) is stopping time, we have (2.3) E [ V ( T a (-b ) )] = E [ V (0)] = 1 Hence (2.4) 1 = P ( X ( T a (-b ) ) = a ) exp {-2 μa } + P ( X ( T a (-b ) ) =-b ) exp {-2 μ (-b ) } . And (2.5) P ( T a < T-b < ∞| X (0) = x ) = P ( X ( T a (-b ) ) = a | X (0) = x ) = P ( X ( T ( a-x )(-b-x ) ) = a-x | X (0) = 0) . 1 2XIXI WANG, CITED FROM THE BOOK “APPLIED STOCHASTIC PROCESS” BY YUANLIE LIN, TSINGHUA UNIVERSITY PRESS Therefore P ( T a < T-b < ∞| X (0) = x ) = 1-exp { 2 μ ( b + x ) } exp {-2 μ ( a-x ) } -exp { 2 μ ( b + x ) } = exp { 2 μb } -exp {-2 μx } exp { 2 μb } -exp {-2 μa } (2.6) The problem gets proved....
View Full Document

This note was uploaded on 10/02/2009 for the course STAT 87528 taught by Professor Pitman,jim during the Spring '09 term at Berkeley.

Page1 / 2

lecture27 - ∞ = 1 Let V t = exp-2 μX t which is a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online