Why Jump Processes Are Usually Right-Continuous with Left Limits
Summary
The document explains why stochastic jump processes are commonly modeled as càdlàg: their paths are right-continuous and have left limits. In a Poisson process, the value changes at the jump time, so the jump is visible in the process value at that time. This convention aligns with an adapted process, whose value at a given time is observable using information available by then.
The discussion contrasts this with predictable quantities such as a point process intensity, which should be determined by the history before a possible event. Such quantities are often represented with left-continuous paths. The choice of path convention therefore depends on the role of the process: realized jumps can be adapted and càdlàg, while their intensity can be predictable and left-continuous. The answers are conceptual rather than a formal treatment, and the document notes that modeling needs can call for other conventions.
Key ideas
- A càdlàg Poisson process records a jump in its value at the jump time.
- Adapted processes can reflect events as they occur, even when those events were not predictable in advance.
- A point process intensity is based on the process history and is typically modeled as predictable.
- Left-continuous paths are useful for predictable quantities, while càdlàg paths suit realized jump processes.
- The appropriate path convention depends on the modeling purpose.
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# Why does jump process has to be Cadlag and not the other way around # Why does jump process has to be Cadlag and not the other way around In all books and references that I have been exposed to, the jump processes have been defined to be Cadlag(right continuous with left limits). But no one has explained why this is the preferable case, why can't it be Caglad? I suspect it has something to do with filtration, but I don't know the exact reasoning. ## Answer by M. Jeunesse (score 4, accepted) https://quant.stackexchange.com/a/27764 I don't know if this is enough. But here is my understanding. Let's imagine a simple process like a Poisson process. It is naturally cadlag, because at the time you jump, you jump. Just before, you have not jumped. Mathematically, if the first jump occurs at $t$, $\forall s<t, N_s=0$ and $N_t=1$. It means that the jump occuring at time $t$ is $t$-measurable (even if it is not predictible). So a cadlag process means that at the time of the jump, you see the process jumping. ## Answer by Marine Galantin (score 2) https://quant.stackexchange.com/a/66428 Perhaps not the answer you are expecting but quoting "An introduction to the theory of point processes: Volume I, Elementary theory and methods. Springer, 2002.", you do not always take càdlàg processes. It really depends on what you want to model. - It makes sense to have non-previsible jumps: this is this idea of càdlàg, since it is continuous from the right; - but in the case of point proceses where you have an underlying intensity, one would want the intensity to be continuous from the left! Because you want the conditional intensity to be defined by its history, not by the point itself. Perhaps some insightful keywords would be: you want the intensity to be "predictable", but processes with jumps to be "adapted".
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