lecture27

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

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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 <

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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....
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## 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.

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

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