Solution for Random Walk with Reflecting Boundary at 0

Question

I'm trying to find the general solution for a Markov chain defined as a

random walk with $a + b + c = 1, P(X_k = -1) = a, P(X_k = 0) = b, P(X_k = 1) = c$

and a reflecting boundary at 0, in other words, $Y_{n+1} = \max\{Y_n+X_{n+1}, 0\}$

The flow balance equation seems to be

$(1-b) \pi_{n} = a\pi_{n+1} + c\pi_{n-1} : n\geq1$

Any help would be appreciated

Answer 1

Using the same solution process from ODEs

Solving the flow balance equations gave an equation of form

$\mu s^2-(1-\gamma)s+\lambda=0$

with the general solution from the roots being

$c_1 (\frac{\pi}{\mu})^i + c_2$

Answer 2

I am not sure what you mean by "general solution". You can construct an embedded process, the walk which never stays in one place, as follows: $\tau_0=0$, $\tau_n=\inf\{k>\tau_{n-1}:\ Y_{k}\ne Y_{k-1}\}$ and $X_k=Y_{\tau_k}$. Thus $X_{k+1}=X_k\pm 1$ for all $k=0,1,2,\dots$.

The transition probabilities for this embedded walk are as follows: $$ X_{k+1}=X_k+\begin{cases} +1, &\text{with probability }\frac{c}{c+a};\\ -1, &\text{with probability }\frac{a}{c+a}. \end{cases} $$ Now, this is a standard random walk with probability going right $p:=\frac{c}{c+a}$ and probability for going left $1-p$, for which pretty much everything is known, see e.g. here.