Let $\xi_t$, $t\geq0$, be a pure birth process, with $P\{\xi_{t+h} = i +1 | \xi_t = i\} = \lambda^ih + o(\lambda)$, as $h \downarrow 0$. At $t=0$, $\xi_0 =1$. Let $\tau = \min\{t ~|~ \xi_t = N\}$. Calculate $E(\tau)$.
I am most comfortable with the discrete-time formalism so I attempted an analogous calculation. The event $\{\tau = r\}$ is equivalent to the event that $\xi_t < N$ for all $t<r$ , but $\xi_t \geq N$ for $t\geq r + h$ for any $h>0$.
Question 1: is the probability of that event $\lambda^{N-1}h + o(h)$?
If it is so, (and I think this is the case, since it is a pure birth process that where $t$ is only increasing) then
$$E(\tau) = 1+ \sum_{i=2}^{\infty} i(\lambda^{N-1}h + o(h)) = 1+ h\sum_{i=2}^{\infty} i(\lambda^{N-1}) + \sum_{i=2}^{\infty}io(h)$$
So $$\frac{1}{h}E(\tau) = \frac{1}{h} + \lambda^{N-1}\sum_{i=2}^{\infty} i + \frac{o(h)}{h}\sum_{i=2}^{\infty}i$$
But I can't take $h\to 0$, as I am running into an infinite sum $\sum_{i=2}^{\infty} i$, which leads me to believe that my original interpretation was wrong. Can anyone point out where I went wrong here and how to work it out?
As in every birth chain, the jump from $i$ to $i+1$ occurs after an exponential time $T_i$ with parameter $t_i$ (and here $t_i=\lambda^i$) hence, starting from $1$, $E_1(\tau)=E(T_1+\cdots+T_{N-1})=1/t_1+\cdots+1/t_{N-1}$.
For every $t$, the probability that $\{\tau=t\}$ is zero.