Investigation and Development of the Methodologies for Simulating Self-similar Processes

TL;DR

This paper reviews existing methods and introduces a new generalized approach for simulating a broad class of self-similar processes using a modified inverse Lamperti transformation, demonstrated on fractional Brownian motions.

Contribution

It provides a comprehensive review and enhancement of existing simulation methods and proposes a novel, general approach for simulating self-similar processes.

Findings

Improved simulation techniques for specific self-similar processes.

A new general method for simulating self-similar processes.

Successful application to fractional and sub-fractional Brownian motions.

Abstract

This paper is devoted to the study of simulating a large class of self-similar processes. Since most current simulation approaches are limited to case-by-case studies, every existing approach has its constraints and flaws; hence a general and efficient simulation approach is in demand. Our study sheds some light in this direction. The paper's contributions are bi-fold. First, reviews and improvements are made to some existing methods for simulating specific self-similar processes. Second, we propose a novel method to simulate a general self-similar process, where we use a modified inverse Lamperti transformation to transform self-similarity to stationarity. Successful applications are made to simulate fractional Brownian motion and sub-fractional Brownian motion.