Compute gambler’s ruin probabilities and hitting times

Q: Compute gambler’s ruin probabilities and hitting times

This question evaluates understanding of stochastic processes, Markov chains, and discrete-time random walks by requiring derivation of absorption probabilities and expected hitting times in the gambler’s ruin model.

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A gambler plays a sequence of independent bets. Starting wealth is $i$ dollars, with absorbing boundaries at $0$ (ruin) and $N$ (target).

Each round:

With probability $p$ , wealth increases by 1.
With probability $q=1-p$ , wealth decreases by 1.

Derive the probability $u_i$ that the gambler reaches $N$ before 0.
Derive the expected number of steps until absorption (hitting either 0 or $N$ ) starting from $i$ .
(Random walk follow-up) For the unbiased case $p=q=1/2$ , relate your results to a simple symmetric random walk hitting $\{0,N\}$ .

Compute gambler’s ruin probabilities and hitting times

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