# Martingale stochastic processes

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.

• 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 $k<n$. Dec 17 '20 at 17:55