Fractional L\'{e}vy-driven Ornstein--Uhlenbeck processes and stochastic differential equations

TL;DR

This paper develops a pathwise integral framework for fractional Lévy processes, introduces a fractional Lévy-Ornstein-Uhlenbeck process, and applies it to solve and analyze fractional SDEs with long-range dependence.

Contribution

It introduces a novel pathwise integral driven by FLPs, defines the FLOUP process, and demonstrates its use in solving fractional SDEs with long-range dependence.

Findings

FLOUP is the unique stationary solution of the Langevin equation.

FLOUP exhibits long-range dependence in its increments.

Solutions to fractional SDEs driven by FLPs are constructed using FLOUP.

Abstract

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.