Strong approximation of continuous local martingales by simple random walks

TL;DR

This paper demonstrates how any continuous local martingale can be approximated almost surely by a sequence of simple symmetric random walks, using a time change based on quadratic variation, extending Brownian motion constructions.

Contribution

It introduces a method to approximate continuous local martingales with simple random walks via quadratic variation, providing convergence rates and conditions for independence.

Findings

Almost sure convergence of random walks to martingales.

Rates of convergence are nearly optimal.

Symmetry of increments characterizes independence conditions.

Abstract

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or, equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone.