Multi-Mixed Fractional Brownian Motions and Orstein-Uhlenbeck Processes

TL;DR

This paper introduces multi-mixed fractional Brownian motions and Ornstein-Uhlenbeck processes, analyzing their existence, path properties, and dependence structures, which are constructed by superimposing infinitely many fractional processes.

Contribution

It establishes the existence of these complex mixed processes and investigates their detailed path properties and dependence characteristics.

Findings

Proved existence as $L^2$ processes.

Analyzed long-range and short-range dependence.

Studied H"older continuity and $p$-variation.

Abstract

We study the so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein--Ulhenbeck (mmfOU) processes. These processes are constructed by mixing by superimposing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein--Uhlenbeck processes (fOU), respectively. We prove their existence as $L^2$ processes and study their path properties, viz. long-range and short-range dependence, H\"older continuity, $p$-variation, and conditional full support.