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

Abstract

Using Riemann-Stieltjes methods for integrators of bounded p-variation we define a pathwise integral driven by a fractional Lévy 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 Lévy-Ornstein-Uhlenbeck process (FLOUP) pathwise as an improper RiemannStieltjes 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 Lévy-driven Cox-Ingersoll-Ross (CIR) models.

Conditional Distributions of Processes Related to Fractional Brownian Motion

“…The existence of this expression for the complex arguments above is ensured by the existence of the left hand side of the equation. We want to emphasize that this solution is in contrast to the classical Brownian case (κ = 0) not unique; for further details we refer to Fink and Klüppelberg [7], Proposition 5.1. We considered there a similar case for fractional Lévy processes.…”

Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk

“…When d ∈ (0, 1 2 ) n , it is also possible to define pathwise integration with respect to MG-FLPs using Hölder continuity, as used in [10], or a p-variation approach; see [16]. Another possible approach would be via a Skorohod-type integral using the S-transform as suggested by Bender and Marquardt [4].…”

“…. However, as already mentioned in [16], this does not cover CIR type processes. , d(n)) ∈ (0, 1 2 ) n , general SDEs driven by MG-FLPs can be considered using, for example, the theory of Zähle [40], Section 5.…”

“…. Using the zero-quadratic variation property and the integration concept of Russo and Vallois [32], a theory similar to that of [16] can be proven (at least up to stationary solutions, where we have to address Remark 4.1). However, as already mentioned in [16], this does not cover CIR type processes.…”

Generalized fractional Lévy processes with fractional Brownian motion limit

“…It is then a natural step to extend an FBM to FLPs providing more flexible distributions and tail behavior than Gaussian processes, retaining the long memory increments. Fink and Klüppelberg (2011). Marquardt (2006)).…”