Conditioning to avoid zero via a class of concave functions for one-dimensional diffusions

TL;DR

This paper introduces a novel conditioning method for one-dimensional diffusions on the half-line to avoid zero, using supermartingales defined via concave functions related to the scale function, and studies the resulting limit laws.

Contribution

It develops a new class of conditioning for diffusions using concave functions and analyzes the associated limit laws, extending existing theory.

Findings

Existence of the conditioning through supermartingales is established.

Absolute continuity relations of the limit laws are characterized.

A framework for avoiding zero in diffusions via concave functions is proposed.

Abstract

For one-dimensional diffusions on the half-line, we study a specific type of conditioning to avoid zero. We introduce supermartingales defined via concave functions with respect to the scale function. A conditioning is formulated through the exit times of the supermartingale, and its existence is shown. We also investigate the absolute continuity relations of the limit laws at time infinity.