Consider a random walker:
A markov process $Y_t:t>0$ takes value from finite space $\{\chi\}_{n\in\mathbb Z^+\&n\le L}$. The probability of jumping from $n\to n-1$ is $$\mathbb P(\chi_{n-1},t+dt|\chi_{n},t)=\rho e^{-L\theta_\beta(\rho)}$$ where $\rho=\frac nL$ and probability of jumping from $n\to n+1$ is $$\mathbb P(\chi_{n+1},t+dt|\chi_{n},t)=(1-\rho) e^{-L\theta_\alpha(\rho)}$$ Above $\theta_{\beta}\,and\,\theta_\alpha$ are some function of $\rho$.
Question What is hitting time, viz., when the walker will hit the wall at $\rho=0$ given it starts from $\rho=\rho_i$?
For finding a closed-form solution, one can consider $$\theta_\alpha(\rho)=\rho(2\alpha-\rho)\,and\,\theta_\beta(\rho)=\rho(2\beta-\rho)$$