I’m quite rusty, and this was never my field, but it isn’t immediately obvious why you get the…
Phyllis Little

So the transition operator in the continuous time case might be thought of as a matrix of differentials. Instead of s mapping to Ts, it instead maps to Tsdt. Then we can integrate over the time domain to get the total change.

I was actually a bit concerned about the notation I used, as it isn’t clear that the probability distribution/density is a function of time.