Strong approximation of continuous local martingales by simple random walks

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 walk, 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 [8].