Applying the forward equation on a conditional expectation of a continuous Markov Chain

Question

Suppose $X$ is a homogeneous,continuous Markov chain taking values in $\mathbb{N_0}$ with $Q$-matrix given by : $q_{n,n+1}=\lambda n + \mu$ and $q_{n,n}=-q_{n,n+1}$. Given the mapping, for $k\in \mathbb{N_0}$ and $t\in \mathbb{R_+}$, $m(t)=\mathbb{E}(X_t|X_0=k)$, I am asked to find an expression for $m(t)’$, by using the forward equation for continuous Markov chain and then solve the resulting differential equation. However, I do not know how to approach the first part at all. A hint, on how to actually apply the forward equation, would be appreciated.

Answer 1

Upon further thinking, I have come up with a solution. By conditioning, we get: $m(t)’=\sum_{n=0}^{\infty}nP’_{k,n}(t)$. By simply evaluating the matrix product of $P(t)’=QP(t)$ in the $(n,k)$-th entry and substituting that expression in, we can get a term for $m(t)’$ in terms of $\lambda$ and $\mu$, that is $m(t)’=\lambda m(t) +\mu$. With this the the first order differential equation is solved easily.