為什麼跳轉過程必須是 Cadlag 而不是相反
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.
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.
也許不是您期望的答案，而是引用*“點過程理論簡介：第一卷，基本理論和方法。斯普林格，2002 年。”* ，您並不總是採用 càdlàg 程序。這實際上取決於您要建模的內容。
- 有不可預知的跳躍是有意義的：這就是 càdlàg 的概念，因為它從右側連續；
- 但是對於具有潛在強度的點過程，人們會希望強度從左側連續！因為您希望條件強度由其歷史定義，而不是由點本身。
也許一些有見地的關鍵詞是：你希望強度是“可預測的”，但跳躍的過程是“適應的”。