0

I'm having some trouble understanding the following theorem: enter image description here

How do you interpret the sentence after the "then", involving the a.s. property for $X_s$ when $s \in [T,\infty[$?

I interpret it as $P(X_s=0 \ \forall_{S \in [T,\infty[})=1$. However, I'm not sure how intuitively this 'connects' with the definition of $T$... Simply looking at the definition of $T$, I don't see how it would make a supermartingale to be $0$ a.s. .

An old man in the sea.
  • 5,241
  • Perhaps you can interpret this as "a nonnegative càdlàg supermartingale that hits zero stops on its own" (i.e. without needing to force stopping through $T$), – Jose Avilez Jun 14 '21 at 21:18
  • @JoseAvilez but what would be the intuition for that? – An old man in the sea. Jun 14 '21 at 21:25
  • 1
    @Anoldmaninthesea. The intuition would be that a supermartingale tends to go down, so if $X_t$ is always non-negative and $X$ hits $0$, there is no way for it to go down any farther so it has to stay at $0$. It's sort of similar to the intuition for why a non-negative random variable with expectation $0$ has to be $0$ a.s.: There is no way for it to cancel out a path that increases back above $0$ with one that decreases below $0$. – user6247850 Jun 15 '21 at 18:50
  • @user6247850 Thanks. I think I get it now. ;) – An old man in the sea. Jun 15 '21 at 18:57

0 Answers0