Gambler's problem

Question

Let N$\geq 2$ be an integer. Consider a gambler who starts with i$<N$ Euro. For each successive gamble the gambler either wins 1 Euro with probability p or loses 1 Euro with probability q=1-p. Denote $X_n$ as the total fortune of the gambler after the $n^{th}$ gamble. The goal is to reach a fortune of N Euros without running out of money. My question is this: How can I compute the probability $\mathbb{P}_i(N)$ that the gambler, starting with i Euros, reaches a fortune of N before ruin, if $p\neq 0.5$? I know that for p=0.5 the probability is $\frac{i}{N}$.

Answer 1

You can use martingale optimal stopping time theorem (OST). The problem is find $\mathbb{P}(\tau_N < \tau_0)$ where $\tau_i = \inf\{n \ge 0 \hspace{0.2cm} \text{such that} \hspace{0.2cm} S_n \in \{0, N\} \}$ and where $S_n = i + \sum_{k=1}^{n} X_k$ (with convention $\sum_{k=1}^{0} X_k = 0$

• If $p \neq q$ you can show that $\left(\left(\frac{q}{p} \right)^{S_n}\right)_{n \ge 0}$ is a martingale. Then applying OST and you get:

$$\mathbb{P}(\tau_N < \tau_0)(\frac{q}{p})^{N} + \mathbb{P}(\tau_N > \tau_0) = (\frac{q}{p})^i$$. Then :

$$\mathbb{P}(\tau_N < \tau_0) = \frac{(\frac{q}{p})^i - 1}{(\frac{q}{p})^N - 1}$$

I did it quickly so feel free to correct if there are mistakes