Fractional Brownian Motion

Appendix 3 Simulation of fBm

In this appendix, we describe several simulation procedures for fBm. We concentrate on the so-called exact methods. They allow us to model a sample from fBm as a Gaussian vector with a certain covariance matrix. Section A3.1 is devoted to the basic method, the Cholesky decomposition of the covariance matrix. Section A3.2 describes a faster algorithm of finding this decomposition. Section A3.3 describes a more efficient method of simulation based on the so-called circulant embedding of the covariance matrix. Section A3.4 gives a brief overview of various approximate simulation methods, which may be convenient and efficient in some particular problems.

A3.1. The Cholesky decomposition method

Suppose that we need to simulate the trajectory of fBm . Practically, it is only possible to simulate in discrete time. Therefore, we take an equidistant partition of the interval [0, T] and consider the following problem: how to simulate the values These values form a centered Gaussian vector with a certain covariance matrix. Therefore, they can be simulated as a linear transform of a standard Gaussian vector (a sequence of independent standard normal random variables). ...