One-Dimensional Random Walk with Reflecting Barrier

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Markov Chains / Stationary Distributions.

Your Task
Consider a discrete-time random walk on the finite state space $S = \{0, 1, \dots, N\}$. From state $i \in \{1, \dots, N-1\}$, the walk moves to $i+1$ with probability $p$ and to $i-1$ with probability $q = 1-p$. The boundaries are reflecting:

• From state $0$, the walk moves to state $1$ with probability $1$.
• From state $N$, the walk moves to state $N-1$ with probability $1$. Derive the stationary distribution $\pi = (\pi_0, \pi_1, \dots, \pi_N)$.

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.