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,
[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.