Limit probability for some Hitting time of a Feller Process

Question

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 :

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

Regards

I think it is more usually called a Feller process – Henry Feb 24 '11 at 14:53 — Henry, Feb 24 '11 at 14:53
@Henry : post edited accordingly. – TheBridge Feb 24 '11 at 18:24 — TheBridge, Feb 24 '11 at 18:24

score 4 · Accepted Answer · answered Feb 24 '11 at 16:15

Some useful facts about Feller processes are at http://almostsure.wordpress.com/2010/07/19/properties-of-feller-processes/.

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.

Limit probability for some Hitting time of a Feller Process

1 Answers1