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Stationary and self-similar processes driven by Lévy processes

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  • Barndorff-Nielsen, Ole E.
  • Pérez-Abreu, Victor

Abstract

Using bivariate Lévy processes, stationary and self-similar processes, with prescribed one-dimensional marginal laws of type G, are constructed. The self-similar processes are obtained from the stationary by the Lamperti transformation. In the case of square integrability the arbitrary spectral distribution of the stationary process can be chosen so that the corresponding self-similar process has second-order stationary increments. The spectral distribution in question, which yields fractional Brownian motion when the driving Lévy process is the bivariate Brownian motion, is shown to possess a density, and an explicit expression for the density is derived.

Suggested Citation

  • Barndorff-Nielsen, Ole E. & Pérez-Abreu, Victor, 1999. "Stationary and self-similar processes driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 357-369, December.
    • Handle: RePEc:eee:spapps:v:84:y:1999:i:2:p:357-369
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    References listed on IDEAS

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    1. Tina Hviid Rydberg, 1997. "A note on the existence of unique equivalent martingale measures in a Markovian setting," Finance and Stochastics, Springer, vol. 1(3), pages 251-257.
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    Cited by:

    1. Wiktorsson, Magnus, 2002. "Simulation of stochastic integrals with respect to Lévy processes of type G," Stochastic Processes and their Applications, Elsevier, vol. 101(1), pages 113-125, September.
    2. Grahovac, Danijel, 2020. "Multifractal processes: Definition, properties and new examples," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    3. Rubenthaler, Sylvain & Wiktorsson, Magnus, 2003. "Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 1-26, November.

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