Optimal Blind Buy Time in a Three-State Process

A stock price evolves through three qualitative states during a trading day: "calm" (state 1), "active" (state 2), and "exhausted" (state 3). The stock starts in state 1 at time $t = 0$. The time spent in state 1 before transitioning to state 2 is $T_1 \sim \text{Uniform}[0, 1]$; independently, the time spent in state 2 before transitioning to state 3 is $T_2 \sim \text{Uniform}[0, 1]$. Once in state 3, the stock remains there until the market close at $t = 2$. You may submit a single market buy order at a deterministic time $\tau \in [0, 2]$. Crucially, you do not observe the current state before trading -- you must commit to $\tau$ in advance. (a) Express the probability that you buy while the stock is in state 2 (the "active" state) as a function of $\tau$. (b) Find the value of $\tau$ that maximizes this probability and compute the maximized probability.