Convergence results for the Time-Changed fractional Ornstein-Uhlenbeck processes

TL;DR

This paper investigates the convergence behavior of time-changed fractional Ornstein-Uhlenbeck processes, showing Gaussian limits, convergence to classical OU processes as the Hurst parameter approaches 1/2, and analyzing related PDE solutions.

Contribution

It provides new convergence results for the distribution and topology of time-changed fractional Ornstein-Uhlenbeck processes, including Gaussian limits and Skorohod topology convergence.

Findings

Gaussian limit distribution for the process despite time change

Convergence to classical Ornstein-Uhlenbeck process as Hurst index approaches 1/2

Convergence in Skorohod J1-topology as H approaches 1/2

Abstract

In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/2^+$ in the space of c\`adl\`ag functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker-Planck equation associated to the aforementioned processes, as $H \to 1/2^+$.