Multifractal random walks with fractional Brownian motion via Malliavin calculus

TL;DR

This paper introduces a new multifractal random walk model driven by fractional Brownian motion, analyzed with Malliavin calculus, and demonstrates its application in finance to replicate empirical data features like leverage effect.

Contribution

It presents a novel stochastic integral construction of multifractal random walks with fractional Brownian motion using Malliavin calculus, and applies it to financial modeling.

Findings

Model captures key empirical features of financial data.

Numerical simulations validate the theoretical properties.

The approach extends existing multifractal models with fractional Brownian motion.

Abstract

We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model in finance that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations.