Central limit theorems for martingales-I : continuous limits

TL;DR

This paper establishes weak convergence results for martingales with continuous limiting compensators, showing they converge to a time-changed Brownian motion and exploring independence conditions, with applications in finance and stochastic processes.

Contribution

It provides new weak convergence theorems for martingales with continuous limits and characterizes conditions for independence between the martingale and its limit.

Findings

Martingales converge to Brownian motion evaluated at the compensator

Sufficient conditions for independence between martingale and limit

Applications to occupation times and financial volatility estimators

Abstract

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. Examples of applications are provided, notably for occupation time processes and statistical estimators of financial volatility measures.