# Expected Growth of General Pure Birth Process

Assume we have a non-explosive Markov Chain $$(X_t)_t$$ in continuous time with Q matrix $$Q(k,l) = \begin{cases} \lambda_k &\text{if } l=k+1\\ -\lambda_k &\text{if } l=k\\ 0 &\text{otherwise} \end{cases}$$ i.e. $$X$$ is a pure birth process. Further we know that $$0<\lambda_i < \lambda_j$$ for all $$i.

What can we say about $$\mathbb{E}[X_t]$$ if $$X$$ starts in one a.s.?

First I tried to calculate the transition function using Kolmogorov's forward equation, but this ends in solving a tedious system of linear ODEs. Is there another way to give an expression for $$\mathbb{E}[X_t]$$?

I am happy about any kind of suggestion or help. Thank you.