Hitting-Time Variance via a Compensated Martingale

Consider a Markov chain on $\{0, 1, 2, 3\}$ with transitions: from state $i$ ($0 < i < 3$), the chain moves to $i+1$ with probability $p = \tfrac{2}{3}$ and to $i-1$ with probability $q = \tfrac{1}{3}$. State $0$ and state $3$ are absorbing.

Let $T = \inf\{n \ge 0 : X_n \in \{0, 3\}\}$.

(a) Define $M_n = X_{n \wedge T} - (p - q)(n \wedge T)$. Verify that $M_n$ is a martingale and use the Optional Stopping Theorem to find $E[T \mid X_0 = 2]$.

(b) Find a second martingale of the form $N_n = (X_{n \wedge T} - (p-q)(n \wedge T))^2 - 4pq(n \wedge T)$ and use it to compute $\operatorname{Var}(T \mid X_0 = 2)$.

(c) Verify your answer for $E[T]$ by first-step analysis.