Let $\{X_t\}_{t\geq 0}$ a pure death process with Q-matrix: $$Q=\begin{pmatrix} 0&0&0&0&\cdots0\\ k& -k &0&0&\cdots0\\ 0&k&-k&0&\cdots 0\\ \vdots&&\ddots&\ddots\\ 0&0&\cdots&k&-k \end{pmatrix}$$ And state space $S=\{0,1,2,\cdots,n\}$, find the probabilities $P(X(t)=i)$ for $i=0,1,...,n$ if $X(0)=n.$
Using Chapman-Kolmogorov's equation I have:
$$P_i(t)=\sum_jP(X(0)=j)P_{j,i}(t)=P_{n,i}(t)$$
But the only way that I know for solve that is by the Backward/Forward equations, How can I find the transition functions efficiently?
Let $N(t)$ be be a Poisson process with rate $k$. Since the death rate is constant, the number of deaths in $(0,t]$ is given by $$n-X(t)=N(t)\wedge n.$$ It follows that $$X(t) = (n-N(t))^+, $$ and so $$\mathbb P(X(t)=j) = \mathbb P((n-N(t))^+=j) =\begin{cases}\mathbb P(N(t)\geqslant n),& j=0\\ \mathbb P(N(t)=n-j),& j=1,\ldots,n\end{cases} $$ with $$\mathbb P(N(t)=i) = \frac{e^{-kt}(kt)^i}{i!}. $$