Consider the random walk on $\{0,1,2,\ldots\}$:
If $X_t\neq 0$, $$ X_{t+1} = \left\{\begin{matrix} X_t + 1 & wp.\quad \frac{\gamma}{2}\\ X_t & wp.\quad 1-\gamma\\ X_t - 1& wp.\quad \frac{\gamma}{2} \end{matrix}\right. $$
Otherwise, we say the random walk dies. Let $T_i$ denote the expected life expectancy if we start at $i$, e.g. $T_0=0$, how can I calculate $T_i,i>0$.
My Try
We have
$$\gamma T_i = 1+ \frac{\gamma}{2}T_{i-1}+ \frac{\gamma}{2}T_{i+1}+(1-\gamma)T_i$$
Seems not working.