I've recently started studying Cont & Tankov's "financial modelling with jump processes". I'm curious as to why that this assumption of the cadlag property (also called RCLL "right continuous with left limits") for price paths is natural for the financial context?
Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for the jump or that a jump is "under way"). Each jump is a surprise, after which we believe there will be no jumps at least for a little while.
I hear it in the trading room all the time: "WTF! I looked away from the screen for a few secs and the price just jumped up[down]!". That is exactly what discrete time monitoring of a c.t. cadlag process is like. Even if you "look very frequently" somehow you never really "see" the jump, only the after jump situation.