2

I am following Brownian motion and Stochastic calculus by Karatzas and Shreve

They give the following definitions with an example:

enter image description here

I do not understand the difference between definition 1.1 and 1.3 and consequently I do not understand why $P[Y_t = X_t ; \forall{t} \ge 0] = 0$ in the example.

Edit: I found this that kind of helps, basically saying "at each time $t$" treats $t$ as a countable set and saying "$\forall{0} \le t < \infty$" treats $t$ as an uncountable set? Actually this still does not make much sense to me, I think it has to do with the order of the propositions.

Monolite
  • 4,603
  • In short: indistinguishable says the trajectories are the same with probability 1. Modification-equivalent says the values at each fixed time are the same with probability 1. When the time set is uncountable, it may be that there is always some time where the trajectories deviate but that this time is continuously distributed, so that its probability to take on any particular value is zero. – Ian Sep 02 '15 at 21:04

0 Answers0