A Weak Convergence Criterion Constructing Changes of Measure

TL;DR

This paper introduces a weak convergence-based criterion for verifying when a nonnegative local martingale is a true martingale, replacing traditional integrability conditions with a tightness condition, with various applications.

Contribution

It presents a novel weak convergence approach that simplifies martingale verification and broadens the scope of measure change techniques in stochastic processes.

Findings

Provides a necessary and sufficient weak convergence criterion.

Replaces integrability conditions with tightness conditions.

Applications include simplified proofs and process characterizations.

Abstract

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes.