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Does anyone know how to do this question?

A player whose initial holding is $N$ bets 1 on each game of a series of independent identical parts. He loses his bet whether he loses or he wins but, if he wins, he gets back $k$ times his stake. It is assumed that the player has a probability $p$ of winning each game independently others. We set $M_{n} = S_{n} − n (kp − 1)$ for $n ≥ 0$, where $S_{n}$ represents what the player has after $n$ games.

(a) Show first that $M_{n}$ for $n ≥ 0$ is a martingale and $T = min \{n ≥ 0: S_{n} = 0\}$ a stopping time with respect to $S_{n}$ for $n ≥ 0$.

(b) determine $E(T)$ in the case where $kp < 1$ assuming that the conditions of the martingale stop theorem are verified in this case with a finite average downtime.

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  • $\begingroup$ To show it is a martingale, find the recursive expression for $S_n$, and especially for $E[S_n]$. Then $M_n$ is a martingale if $E[M_n|M_k] = M_n$ for some $k<n$. $\endgroup$ – Pontus Hultkrantz Dec 17 '20 at 17:55

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