Probability of a process being zero

I found proofs in books (for example Oksendal's) where there is a process $$S_t$$ which satisfies $$\mathrm E[S_t^2]=0$$ for all $$t$$. Based on this, a conclusion is made that

$$\mathbb P[S_t = 0 \textrm{ for all } t \in \mathbb Q \cap [0,T]]=1$$ for some $$T>0$$ and where $$\mathbb Q$$ is the set of rational numbers. Then, using the continuity of $$S_t$$ a further conclusion is made that $$\mathbb P[S_t = 0 \textrm{ for all } t \in [0,T]] = 1$$

My question is, why can one conclude the first probability but not the second from $$\mathbb E[S_t^2]=0$$? Also, probably related, how do I show these conclusions rigorously?

• Regarding your first question: $E[S_t^2]=0\Rightarrow P(S_t=0)=1$ is a standard result for random-variables, but you cannot conclude $\forall t\leq T:P(S_t=0)=1\Rightarrow P(\forall t\leq T:S_t=0)$. (Think for example of a process on $[0,1]$ with a jump at a random point $U\in\mathcal{U}([0,1])$.) But at least, now you can argue with $\sigma$-continuity arguments, because you are dealing with a countable union. This is not possible for the second probability. Nov 10, 2020 at 11:10
• The second probability should follow by the fact, that your process is $\omega$-wise continuous, so if for allmost-all $\omega$ the realizations are constant on a dense subset of $[0,T]$, they must be constant on $[0,T]$. Nov 10, 2020 at 11:16
• Thanks! For the first part, can you give some more references on those "$\sigma$-continuity arguments"? Nov 10, 2020 at 11:23
• Sorry, I think $\sigma$-additivity makes more sense. Let $(A_n)$ be a countable family of measurable sets with probability 1. You can argue $1\geq P(\cap_n A_n)=1-P(\cup_n A_n^c)\geq 1- \sum_nP(A_n^c)=1$. Nov 10, 2020 at 11:30

1 Answer

For a single $$t$$, we have $$E(S_t^2)=0$$ implies $$P(S_t\neq 0)=0$$. Now if $$t_1,t_2,\ldots$$ are any countable set of times, the probability that $$S_{t_i}\neq 0$$ for some $$i$$ is at most the sum of the individual probabilities, i.e. $$0$$.

But we cannot in general make the same conclusion for uncountable sets of times; for example if $$S_t$$ is the indicator function of a Poisson point process then we have $$E(S_t^2=0)$$ for any given $$t$$, yet almost surely there are infinitely many values of $$t$$ with $$S_t=1$$.