Measure changes with extinction

TL;DR

This paper investigates the behavior of measure changes via martingales, clarifying conditions under which the martingale's limit and hitting time coincide, with implications for stochastic process theory.

Contribution

It provides a necessary and sufficient condition linking the limit of a martingale to its hitting time under a changed measure.

Findings

$1/Z_t$ is generally a supermartingale, not a martingale, under the changed measure.

A precise condition is established for the event that the martingale's limit is zero to match the event that it hits zero in finite time.

The results clarify the relationship between martingale limits and hitting times in measure change scenarios.

Abstract

We consider a change of measure by a martingale $Z_t$ and clarify that in general $1/Z_t$ is only a supermartingale under the changed measure. We then give a necessary and sufficient condition for the event that the limit of the martingale is zero to coincide with the event that the martingale hits zero in finite time (up to a set of zero probability).