ONE-DIMENSIONAL RANDOM WALKS
Solve the gambler’s ruin problem for p q random walk by setting up and solving a difference equation. (Reformulate the difference equation as a matrix equation, and use this to represent the solution as a matrix multiplication. To get a simple formula …
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galton.uchicago.eduMany stochastic processes behave, at least for long stretches of time, like random walks with small but frequent jumps. The argument above suggests that such processes will look, at least approximately, and on the appropriate time scale, like Brownian motion. Second, it suggests that many important “statistics” of the random walk will have lim-
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1 Gambler’s Ruin Problem - Columbia University
www.columbia.edu1 Gambler’s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1−p respectively. Let R n denote the total fortune after the nth gamble. The gambler’s objective is to reach a total