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Stochastic processes: martingales

  1. Nov 17, 2008 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    3. 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:
    [tex]T= \sum X_I [/tex]

    But how must I proceed?
  2. jcsd
  3. Nov 17, 2008 #2
    Let [tex]Y_{n}[/tex] be the gambler's winnings after n games. Clearly, [tex]Y_{n}[/tex] is a martingale. We introduce a new stochastic process [tex]Z_{n}[/tex], where [tex]Z_{n}={Y_{n}}^2-n[/tex]. It can be shown that [tex]Z_{n}[/tex] is a martingale with respect to [tex]Y_{n}[/tex]. (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 [tex]E(Z_{N})=E({Y_{N}}^2)-E(N)[/tex]. By applying the Martingale Stopping Theorem (first check the necessary conditions are satisfied), we can show [tex]E(Z_{N})=0[/tex].

    This leaves us with [tex]E(N)=E({Y_{N}}^2)[/tex]. To determine [tex]E({Y_{N}}^2)[/tex], use the definition of expectation, and observe that [tex]Y_{N}[/tex] 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 [tex]E({Y_{N}}^2)[/tex], which will be equal to [tex]E(N)[/tex].
    Last edited: Nov 18, 2008
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