Journal of Theoretical Probability

, Volume 31, Issue 1, pp 445–465 | Cite as

The Multifractal Random Walk as Pathwise Stochastic Integral: Construction and Simulation

  • Soledad Torres
  • Ciprian A. Tudor


We define a multifractal random walk (MRW) as an anticipating pathwise integral, as limit of Riemann sums. The MRW is usually defined as the limit as \(r\rightarrow 0\) of the family of stochastic processes \((X_{r})_{r>0}\) where
$$\begin{aligned} X_{r}(t)=\int _{0} ^ {t} Q_{r}(u)\hbox {d}W(u), \quad t\ge 0, \end{aligned}$$
where W is a Wiener process and Q an infinitely divisible cascading noise (IDC noise) not adapted to the filtration generated by W. In order to define the stochastic integral \(X_{r}(t)\) and to simulate it, one usually assumes that Q and W are independent. Our purpose is to define the MRW with a dependence structure between the IDC noise Q and the Wiener process W. Our construction is done by using Riemann sums, and it allows the simulation of the process.


Malliavin calculus Multifractal random walk Pathwise integration Scaling Infinitely divisible cascades Skorohod integral 

Mathematics Subject Classification (2010)

60C30 60H07 60H05 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CIMFAV, Facultad de IngenieríaUniversidad de ValparaísoValparaisoChile
  2. 2.Laboratoire Paul PainlevéUniversité de Lille 1Villeneuve d’AscqFrance

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