Let be a two dimensional standard Brownian motion with respect to a filtered probability space , starting from a point .

It is well known that 2-D Brownian motion is neighborhood recurrent, but point polar. In particular, following statement holds: for any given point

Can we extend this fact to continuous non-degenerate martingale of the form below?

under uniform non-degenerate condition? More precisely,

Assumption 1is a -progressively measurable 2 by 2 matrix process satisfying

[**Q**] With Assumption 1, can we prove

**So far, we do not have the answer. **

At this stage, we can prove that (1) is true if is further assumed piecewise constant, that is

[**Claim**] Suppose satisfies Assumption 1 and admits the form of

for some partition and . Then, (1) is true.

*Proof:* It based on the fact on point polar property of 2-D Brwonian motion, and induction.

Next, the original attempt to answer [**Q**] was the following: assuming non-piecewise constant of Asssumption 1

- Let be a piecewise constant generate by with uniform -equal-mesh:
- Let be
Note that, converges to in distribution by the definition of Ito integral, denoted by .

- By [
**Claim**] above, we know

However, it is not sufficient to draw the conclusion of (1) from the facts and , by Portmanteau theorem. We tried to prove the set of processes hitting is a -continuity set in some process space under some appropriate topology, but we failed to cover this gap.

Any suggestion will be greatly appreciated.

Hi. As this post is quite old now, maybe you have worked out a solution. However, I can answer the question – no, statement Q is false.

To construct a counterexample, it is more convenient to move the point (1,1) to the origin, so the question is whether with can ever hit the origin with positive probability. To construct a counterexample, set

where and is equal to U with a 90 degree rotation applied (i.e., it is a unit vector orthogonal to Y). Then, is a stochastic differential equation. As is a smooth function of Y (for ), it has a unique solution up until the first time Y hits 0 (once Y hits zero, we have the counterexample, so don’t really care any more and can take to be the identity if we like). Your condition is also satisfied with and . Setting it can be seen that this satisfies the SDE

This is a process, which hits zero eventually with probability 1, and hits zero by any time t with positive probability, so long as .

Dear George, so grateful for your answer and you really understand the subject. Indeed, I did not have counter-example, and always believed it was true.

Fixed, by the way, what is in your definition of ?

My X should be a Y. Also, the b inside the max and min should be squared (and you may as well delete my comments pointing out typos once they’re fixed.

Dear George, Thanks. I will fix it and delete the comments after.

Dear George, After my calculation, I got different SDE for , that is

I am sorry, but would you verify once more and confirm it?

I think I am correct. Itos formula gives . Then use . So the first term on the rhs has quadratic variation , and you can take it from there.

Dear George, Yes you are right. I made a very simple mistake.

Dear George, Yes you are right. I made a very simple mistake.