Advertisement

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
Article
  • 167 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

60C30 60H07 60H05 

References

  1. 1.
    Abry, P., Chainais, P., Coutin, L., Pipiras, V.: Multifractal random walks as fractional Wiener integrals. IEEE Trans Inf Theory 55(8), 3825–3840 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bacry, E., Duvernet, L., Muzy, J.F.: Continuous-time skewed multifractal processes as a model for financial returns. J Appl Probab 49, 482–502 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bacry, E., Muzy, J.-F.: Multifractal stationary random measures and multifractal walks with log-infinitely divisible scaling laws. Phys Rev E 66, 056121 (2002)CrossRefGoogle Scholar
  4. 4.
    Bacry, E., Kozhemyak, A., Muzy, J.-F.: Continuous cascade models for asset returns. J Econ Dyn Control 32(1), 156–199 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Calif, R., Schmitt, F.: Modeling of atmospheric wind speed using a lognormal stochastic equation. J Wind Eng Ind Aerodyn 109, 1–8 (2012)CrossRefGoogle Scholar
  6. 6.
    Calif, R., Schmitt, F., Huang, Y.: Multifractal description of wind power fluctuations using arbitrary order Hilbert spectral analysis. Phys. A 392, 4106–4120 (2013)CrossRefGoogle Scholar
  7. 7.
    Calvet, L., Fisher, A.: Multifractal volatility: theory, forecasting and princing. Cambridge University Press, Cambridge (2008)Google Scholar
  8. 8.
    Chainais, P., Riedi, R., Abry, P.: On non-scale invariant infinitely divisible cascades. IEEE Trans Inf Theory 51(3), 1063–1083 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fauth, A., Tudor, C.A.: Multifractal random walks with fractional Brownian motion via Malliavin calculus. IEEE Trans Inf Theory 60(3), 1963–1975 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fisher A, Calvet L, Mandelbrot B (1997) Multifractality of Deutschemark/US dollar exchange rates. Cowles Foundation Discussion Paper No. 1166Google Scholar
  11. 11.
    Huang, Y., Schmitt, F., Lu, Z., Liu, Y.: Analysis of daily river flow fluctuations using empirical mode mode decomposition and arbitrary order Hilbert spectral analysis. J Hydrol 373, 103–111 (2009)CrossRefGoogle Scholar
  12. 12.
    Mandelbrot, B.: The variation of certain speculative prices. J Bus 36, 394 (1963)CrossRefGoogle Scholar
  13. 13.
    Muzy, J.F., Delour, J., Bacry, E.: Modeling fluctuations of financial time series: from cascade process to stochastic volatility model. Euro Phys J B 17, 537–548 (2000)CrossRefGoogle Scholar
  14. 14.
    Schertzer, D., Lovejoy, S.: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J Geophys Res 92(D8), 9693–9714 (1987)CrossRefGoogle Scholar

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

Personalised recommendations