Fractional Lévy-driven Ornstein--Uhlenbeck processes and stochastic differential equations

Details Fractional Lévy-driven Ornstein--Uhlenbeck processes and stochastic differential equations Fractional Lévy-driven Ornstein--Uhlenbeck processes and stochastic differential equations

2011-02-09

arxiv.org/abs/1102.1830v1

Bernoulli 2011, Vol. 17, No. 1, 484-506

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ281 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ281

Using Riemann-Stieltjes methods for integrators of bounded $p$-variation we define a pathwise integral driven by a fractional L\'{e}vy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we introduce an Ornstein-Uhlenbeck model by a stochastic integral representation, where the driving stochastic process is an FLP. To achieve the convergence of improper integrals, the long-time behavior of FLPs is derived. This is sufficient to define the fractional L\'{e}vy-Ornstein-Uhlenbeck process (FLOUP) pathwise as an improper Riemann-Stieltjes integral. We show further that the FLOUP is the unique stationary solution of the corresponding Langevin equation. Furthermore, we calculate the autocovariance function and prove that its increments exhibit long-range dependence. Exploiting the Langevin equation, we consider SDEs driven by FLPs of bounded $p$-variation for $p<2$ and construct solutions using the corresponding FLOUP. Finally, we consider examples of such SDEs, including various state space transforms of the FLOUP and also fractional L\'{e}vy-driven Cox-Ingersoll-Ross (CIR) models.

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