If the jump process has a jump rate $\lambda$, is there a process such that $$\mathbb{P}\left(N(t+h)=n+m|\hbox{ }N(t) =n\right) =\begin{cases} 1 - \lambda h + o(h) & \text{if }m=0\\ \lambda h + o(h) &\text{if } m=1,\\ 0 & \text{if } m>1,\\ \end{cases} $$, where right here, $m>1$ is not $o(h)$, but 0? So extending to any large interval, $m>1$ scenario cannot occur, i.e. no two or more jumps.
Is there a jump process that in any finite interval, there can only be maximally one jump?
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probability
probability-theory
random-variables
random
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0Interesting question, will let someone with more knowledge about this provide an answer, my thoughts though would be that by viewing deterministic process as a subset of random processes then this could be made true? And that to be a process with interesting random component it needs the "flexibility" of allowing small, but non-zero, probability of more jumps ? – 2017-02-24
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0Any stopped Lévy process $X_t^\tau$ where $\tau$ is the time of the first jump – 2017-02-24