Consider a random walk $S_n=S_0+\sum^b_{i=1}X_i$ with i.i.d steps $X_i$ taking value $4$ and $-7$ with probabilities $\frac{7}{11}$ and $\frac{4}{11}$ respectively.
I would like to find a constant $\gamma$ such that $Y_n=S^2_n-\gamma n$ is a martingale, and hence to determine the expected duration of the walk until absorption at either boundary.
My attempt:
To impose the martingale condition on $Y_n$, one has to evaluate $$E[Y_{n+1}|\mathcal{F}_{n}]=E[(S_n+X_{n+1})^2-\gamma(n+1)|\mathcal{F}_n]=S^2_n+E[X^2]-\gamma(n+1)\Leftrightarrow \\E[X^2]-\gamma=0$$
We have that $$E[X^2]=16\left(\frac{7}{11}\right)+49\left(\frac{4}{11}\right)=10.18+17.81\approx28$$
So, $\gamma=28$
Is this correct?
How do I determine the expected duration of the walk until absorption at either boundary?
Best Answer
As mentioned in comments, your method for computing $\gamma$ is correct, but there is little hope of getting an exact value for $E[T]$ due to overshoot issues. The idea is to get bounds.
You have $Y_0=E[Y_T]= E[S_T^2] - \gamma E[T]$. Assuming $Y_0$ and $\gamma$ are known, you just need upper and lower bounds on $E[S_T^2]$, which are relatively easy to get in terms of the probability $p$ of first crossing the right threshold. You can get upper and lower bounds on $p$ from your previous question (which used $S_0=E[S_T]$). That question is here: Probability the walk terminates