Distribution of a process dependent on a Markov chain's states

Question

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$

and transition probabilities

$$ \begin{array}{l} \mathbf{P}(X_t=1|X_{t-1}=1)=p_a \\ \mathbf{P}(X_t=0|X_{t-1}=1)=1-p_a \\ \mathbf{P}(X_t=0|X_{t-1}=0)=p_b \\ \mathbf{P}(X_t=1|X_{t-1}=0)=1-p_b \end{array} $$

Now consider the process $$ Y_n = \sum_{t=0}^n X_t $$

I'm interested in computing the distribution of $Y_n$ and $\max_{0\leq t \leq n} Y_t$.

Of course I could simulate the chain, but are there closed form expressions for those distributions?

Answer 1

Elaborating on @Did's comment (letting $a_0(s)=\lambda s$ and $b_0(s)=1-\lambda$ as according to the initial distribution), we have the recurrences \begin{align} a_{n+1}(s) &= p_asa_n(s) + (1-p_b)sb_n(s),\\ b_{n+1}(s) &= (1-p_a)a_n(s) + p_b b_n(s). \end{align} Adding these yields $$a_{n+1}(s) + b_{n+1}(s) = (1-p_a+p_as)a_n(s) + (1-p_b+p_bs)b_n(s), $$ so $$a_{n+1}(s) + b_{n+1}(s) = (1-p_a+p_as)^n\lambda s + (1-p_b+p_bs)^n(1-\lambda).$$ It follows that $$Y_n\stackrel d= \lambda(B_{n,a}+1) + (1-\lambda)B_{n,b} $$ where $B_{n,a}\sim\operatorname{Bin}(n,p_a)$ and $B_{n,b}\sim\operatorname{Bin}(n,p_b)$.