2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin.

I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. with $P(\xi_i=\pm 1)=\frac{1}{2}$. Let $\tau_0 = \inf{\{n:X_n = 0}\}$, $\tau_{x+y} = \inf{\{n:X_n = x+y}\}$, and $\tau=\tau_0\wedge \tau_{x+y}$, which are all stopping times w.r.t. the martingale $(X_n)$.

I know that $E[\tau]=xy$, and $E[X_\tau]=x$.

Am i able to find $E[\tau^2]$ in this case? Any suggestion is welcome! Thanks!

  • $\begingroup$ You should be able to modify this to answer your question: math.stackexchange.com/questions/75969/… $\endgroup$ – user940 Mar 20 '13 at 2:28
  • $\begingroup$ Thanks for your reference. May I know how we can find $E[{T}{S_T^2}]$ for that suggestion? $\endgroup$ – freak_warrior Mar 20 '13 at 2:57
  • $\begingroup$ Nevermind, i think i know the answer, $E[{T}{S_T^2}] = (x+y)E[{T}{S_T}]$ in my case, as $S_T$ only takes values $x+y$ or $0$. $\endgroup$ – freak_warrior Mar 20 '13 at 5:48

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