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We introduce a Multifractal Random Walk (MRW) defined as a stochasticintegral of an infinitely divisible noise with respect to a dependentfractional Brownian motion. Using the techniques of the Malliavin calculus, westudy the existence of this object and its properties. We then propose acontinuous time model in finance that captures the main properties observed inthe empirical data, including the leverage effect. We illustrate our result bynumerical simulations.

Multifractal random walks with fractional Brownian motion via Malliavin calculus