Let $P_t$ be a time series such that $P_{t+1} = \alpha P_t+S_{t+1}$, where $\forall t\geq0 : S_t \sim N(0,\sigma)$

Consider the following game: In each round $t$, a player sees $P_t$ and decides whether to bet or not. Once the player chooses to bet he cannot withdraw his bet, nor can he bet again in later round.

The payoff of a bet that was placed at the $n$-th round is $P_n*sign(P_N)$, where $N$ is the last round of the game.

For $P_0 = 0,N=100,\alpha=0.8$ and $\sigma=1$ what is the optimal strategy?

To be honest, I'm straggling to solve an easier relaxation of this problem: the case where $P_0 = 0,N=2,\alpha=1$ and $\sigma=1$.

Any direction?




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