On simultaneous hitting of membranes by two skew Brownian motions

TL;DR

This paper investigates the conditions under which two dependent skew Brownian motions with membranes at zero either meet or do not meet, establishing criteria for almost sure non-meeting or certain meeting.

Contribution

It provides necessary and sufficient conditions for the meeting of two dependent skew Brownian motions with membranes, including the probability being zero or one.

Findings

Processes almost surely do not meet at fixed points except at membranes.

Conditions for the processes to meet are explicitly characterized.

Probability of meeting is either zero or one.

Abstract

We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.