A class of multifractal processes constructed using an embedded branching process

TL;DR

This paper introduces a new class of multifractal processes constructed via embedded branching processes, enabling efficient simulation and including Brownian motion with multifractal time-change.

Contribution

It develops a novel construction of multifractal processes using multitype branching random walks and provides a Markov representation for efficient simulation.

Findings

Includes Brownian motion with multifractal time-change

Provides a finite Markov representation for simulation

Offers an O(log n) complexity algorithm for process generation

Abstract

We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton-Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step n, we can generate step n+1 in O(log n) operations. Detailed pseudo-code for this algorithm is provided.