Lec8[1] - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 8: First passage and occupation times for random walk Lecturer: Jim Pitman 1 First passage and occupation times for random walk Gambler’s ruin problem on { 0 , 1 ,...,b } with P ( a,a - 1) = P ( a,a + 1) = 1 2 and 0 and b absorbing. As defined in last class, T 0 ,b = first n : X n ∈ { 0 ,b } m a,b := E a ( T 0 ,b ) m 0 ,b = m b,b = 0 Solve this system of equations m a,b = 1 + 1 2 ( m a +1 ,b + m a - 1 ,b ). Recall harmonic equation h ( a ) = 1 2 h ( a + 1) + 1 2 h ( a - 1): a-1 a a+1 h(a-1) h(a+1) Now graphically h ( a ) = 1 + 1 2 h ( a + 1) + 1 2 h ( a - 1) looks like this: a concave function whose value at a exceeds by 1 the average of values at a ± 1 1
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Lecture 8: First passage and occupation times for random walk 2 a-1 a a+1 a+2 h(a-1) h(a) h(a+1) h(a+2) Simplify the main equation to ( m a +1 ,b + m a - 1 ,b ) / 2 - m a,b = - 1 which shows that a m a,b is a concave function. The simplest concave function is a parabola, that is a quadratic function of
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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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Lec8[1] - Stat 150 Stochastic Processes Spring 2009 Lecture...

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