I want to find an optimal stopping rule for the game described below:
Setting:
Let's say I have a discrete time-series $\bf {P_t}$, where $t\in \{0, 1,2,\dots, N\}$,
and $\bf P_0 = 0$,
$\bf P_{i+1} = \begin{cases} \bf P_{i} +1 & \text{with prob.} & \alpha_i \\ \bf P_{i} -1 & \text{with prob.} & 1- \alpha_i \end{cases}$
Note that $\alpha_i$ is only known at time $i$ and not before it. In addition, we know the Probability Mass Function of $\bf P_K$ $\forall k \in \{1,2,\dots N\}$ at time $0$.
Rule:
At any given time $\tau$, I can decide to take $P_\tau$ or continue the game. If at time $N$ I have not taken a value, $P_N$ is automatically given to me.
Goal:
Choose an optimal stopping rule to maximize the game's expected value.
Edit:
My rudimentary attempt has been to choose $\tau$ based on the value that maximizes $P_k$.