I have troubles seeing the following. Consider the classical gambler's ruin problem, betting 1 at each time $t\in \mathbb{N}$, and losing or winning -1 respectively +1 at each time till the fortune of the gambler ends or reaches $n>0$. Define the stopping time $\tau_=\inf\{l\geq 0:X_l\in\{0,n\}\}$, where $X_n$ represent the fortune of the gambler at time $n$. I know already that $\mathbb{P}[X_\tau=n\mid X_0=k]=P_k[X_\tau=n]=k/n$, and I want to prove that for $k\in\{1,...,n-1\}$: $E_k[\tau]=1/2(1+E_{k+1}[\tau])+1/2(1+E_{k-1}[\tau])$, which makes me problems.
I tried the following, but the $+1$ is still a problem.
$E_k[\tau]=1/2E_k[\tau\mid X_1=k+1]+1/2E_k[\tau\mid X_1=k-1]$. For now consider only the first term.
$E_k[\tau\mid X_1=k+1]=E[E[(\tau\mid X_1=k+1)\mid X_0=k]]$. Now considering that $\{X_0=k\}\subseteq\{X_1=k+1\}$ and using the tower property, we get
$E[E[(\tau\mid X_1=k+1)\mid X_0=k]]=E[E[(\inf_{t\geq 0}{\{X_t=n\}}\mid X_1=k+1)\mid X_0=k]]$ and noting that $X_0\neq0$, $\tau=\inf_{t\geq0}{\{X_t=n\}}=\inf_{t\geq1}{\{X_t=n\}}=1+\inf_{t\geq0}{\{X_t=n\}}=1+\tau$.
So $E[E[(\inf_{t\geq 0}{\{X_t=n\}}\mid X_1=k+1)\mid X_0=k]]=E[E[1+\tau\mid X_1=k+1)\mid X_0=k]]=E_{k+1}[\tau]+1$.
Can somebody tell me if I am right? Thank you