I just wonder if there is any model about the optimal path of multiple random walks.
To be more specific, consider a decision maker with initial state $X_0=0$. The decision maker wants to hit either $X=-5$ or $X=15$ as fast as possible. He can choose between two possible random walks in each period: either choose random walk 1: with probability 0.8 $X_{t+1}=X_t-2$ and with probability $X_{t+1}=X_t+8$, or choose random walk 2: with probability 0.2 $X_{t+1}=X_t+8$ and with probability 0.8 $X_{t+1}=X_t-2$. The decision maker can change his random walk in each period. What is the optimal path?
For example, choose RW1 in T=1, if $X_{t+1}=X_t-2$ then choose RW2 in T=2; if $X_{t+1}=X_t+8$ then choose RW1 still in T=2, and.......
Please let me know if there exists any literature related to this question. Thanks!