Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I wanted to know if it was true that if we are given a one-dimensional Feller process taking values in $\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in A\}$ with $A$ a open set (this to avoid measurability complexities but I would be also interested in the general case)

then if $X_0\in \bar{A}^c$ we have :

$\lim_{t\to 0}P[\tau_A<t]=0$


share|cite|improve this question
I think it is more usually called a Feller process – Henry Feb 24 '11 at 14:53
@Henry : post edited accordingly. – TheBridge Feb 24 '11 at 18:24
up vote 4 down vote accepted

Some useful facts about Feller processes are at

In particular, every Feller process admits a cadlag modification. If you are willing to assume you are dealing with such a modification, then your question becomes pretty easy, since by continuity of probability $\lim_{t \to 0} P(\tau_A < t) = P(\tau_A = 0)$. But $\bar{A}^c$ is open and $X_t$ is a.s. continuous at 0 so this has probability 0. It doesn't much matter that $A$ be open or even measurable, since if you replace $A$ by $\bar{A}$ the answer is the same.

If you insist on using a non-cadlag modification, I am not sure that you can say much.

share|cite|improve this answer
Ok I Got it Nate Thanks – TheBridge Feb 25 '11 at 13:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.