On pathwise rate conservation for a class of semi-martingales

In this paper, we generalize an earlier result on pathwise rate conservation for cadlag processes to include a diffusion component. This leads to the occurence of an additional term corresponding to the local time of the process when considering the level crossing formula. This extension serves to show that rate conservation is a pathwise property of a cadlag process subject to it satisfying an o(t) growth condition almost surely. When specialized to the stationary case, we obtain a characterization of the invariant distribution of semi-martingales. We then illustrate the application of the conservation law to obtain the invariant distribution of a reflected Ornstein-Uhlenbeck process.