Doubt concerning Stochastic continuity

Question

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity does not imply continuity of the sample paths. For example a Levy jump process is stochastically continuous. My doubt is that if I think about this definition it means that for any $\epsilon>0 $ and $\forall a >0$ we can find a $t$ close enough to $s$ such that $$P(|X(t)-X(s)|>a)<\epsilon $$ That is the proportion of sample paths where this can happen has a probability less than $\epsilon$. But then the jumps of size greater than $a$ can be allowed only on a small proportion of the sample paths .Then why a Levy jump diffusion process can have jumps on almost every path? This has been bothering me a lot. I am pretty sure my understanding here is very flawed. Thank for your help.

Answer 1

Im not sure whether I fully understood what you are asking, but note that the limit you have must necessarily be from the right (i.e $\lim s \downarrow t$), so the paths are almost surely right-continuous. So how can the process have jumps on almost every path despite the condition? Well, look at for example the NIG process which has jumps at every instant but still has right-continuous paths by definition. Maybe the right-continuity is what is confusing you.