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 $i<j$.
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.