I need a characterization of a probability space wherein the probability measure is changing.

I am not a mathematician, and do not know about stochastic processes, but I've been working through these notes: https://www.math.ku.edu/~nualart/StochasticCalculus.pdf

On page 11, there is a statement about the family of probability measures. I've inserted this as an image below. I'm stuck here; is $P_{t_1...t_n}$ one distribution, or is it a sequence of probability measures?

Sorry if this is really basic.

enter image description here

  • $\begingroup$ This is not that basic. More importantly, can you describe what application you are trying to serve by looking at a changing probability measure? $\endgroup$
    – avs
    Commented Dec 5, 2016 at 23:10
  • $\begingroup$ Those notes look handy for dusting away cobwebs, don't mind if I do :) $\endgroup$
    – Mehness
    Commented Dec 5, 2016 at 23:29
  • $\begingroup$ I don't have a concrete application, but I could invent one; I'll probably ask this as a separate question and link it here. $\endgroup$ Commented Dec 5, 2016 at 23:30

1 Answer 1


$P_{t_1,\ldots,t_n}$ is one distribution, it is the joint distribution $$ \mathbb{P} (X(t_1) \in B_1, \ldots , X(t_n) \in B_n ), $$ where $B_i$ are sets in the relevant $\sigma$-algebra ($\sigma$-field in the notes you linked). The quite remarkable thing that Kolmogorov did here is show that by specifying these probabilities for finite collections of times $(t_1,\ldots,t_n)$ we can extend this to the distribution of realisations of the whole path of the process $X(t) = X(t,\omega)$.

  • $\begingroup$ Hi there - let's just say it's been well over 15 years since have looked at this stuff, and a bunch of my books are in storage! Would I find a proof of this in sth like Rogers & Williams? Any references? Think I have Oksendal / Karatzas and Shreve stashed away somewhere too, thx $\endgroup$
    – Mehness
    Commented Dec 5, 2016 at 23:28
  • $\begingroup$ Yeah Rogers and Williams would certainly have it, I don't have it to hand to check though, I do have Probability by Leo Breiman and it also in there. Any book aiming to discuss stochastic processes will likely have it. $\endgroup$
    – Nadiels
    Commented Dec 5, 2016 at 23:34
  • $\begingroup$ Thx a lot, will go digging $\endgroup$
    – Mehness
    Commented Dec 5, 2016 at 23:37
  • $\begingroup$ Just had a look at Breiman's book contents / sample too, looks really nice $\endgroup$
    – Mehness
    Commented Dec 5, 2016 at 23:43

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