Stochastic processes: martingales

[dirk_mec1]

Homework Statement

http://img411.imageshack.us/img411/4274/50122514bc3.png

Homework Equations

http://img133.imageshack.us/img133/4624/68596500xm4.png

The Attempt at a Solution

I don't know how to start I've found this:

Let X be the the winnings per bet and let the total profit be T then:
$$T= \sum X_I$$

But how must I proceed?

 

[pizzasky]

Let $$Y_{n}$$ be the gambler's winnings after n games. Clearly, $$Y_{n}$$ is a martingale. We introduce a new stochastic process $$Z_{n}$$, where $$Z_{n}={Y_{n}}^2-n$$. It can be shown that $$Z_{n}$$ is a martingale with respect to $$Y_{n}$$. (Can you try to show this?)

Let N be the random variable for the step where the gambler's winnings first reach A or -B. Then, we have $$E(Z_{N})=E({Y_{N}}^2)-E(N)$$. By applying the Martingale Stopping Theorem (first check the necessary conditions are satisfied), we can show $$E(Z_{N})=0$$.

This leaves us with $$E(N)=E({Y_{N}}^2)$$. To determine $$E({Y_{N}}^2)$$, use the definition of expectation, and observe that $$Y_{N}$$ can only take the values A or -B. To calculate the relevant probabilities, apply some formulae related to stopping times of Markov Chains (with stationary transition probabilities). We are now able to compute $$E({Y_{N}}^2)$$, which will be equal to $$E(N)$$.