I am having trouble computing the infinitesimal generator of a rather simple process.
Let us consider a distribution $M$ taking values in the whole half line $\mathbb{R}_+$, and a process $(X_t)_t$ defined by the following dynamic :
- If $X_t = \alpha > 0$, then $X_{t + h} = \alpha - h$ for $0 < h < \alpha$.
- When the process reaches 0, (say at time $\tau$), we set $X_{\tau} = Y_i$, where $(Y_i)_i$ is an i.i.d. sequence of random variables following the distribution $M$, independent from everything else.
Hence, this is a toy example of a jump process in continuous time, but the jumps are somewhat deterministic (at a given time $t$, I know when the next jump will occur). I have no problem computin the infinitesimal generator of the process when $t > 0$, however I do not understand how to obtain the generator around the point 0.
For instance if I chose to make the jump happen exactly at time 0, it seems to me that I will end up with something similar to $$ \lim \limits_{t \to 0} \frac{1}{t} (\mathbb{E}[Y_1]-t) $$ which is certainly not finite.
I found several points about this case in the discrete setting, but this point 0 issue is not relevant in such situations.
Edit: In this discret setting, the jump occurs in a way that we never reach 0, the chain jumping from 1 to the value given by the next jump. This is the behavior which I find problematic in the continuous-time, $\mathbb{R}$ setting, since it means describing the transition probability at some limit point $0^+$.