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Numerical simulation of Generalized Hermite Processes

Antoine Ayache, Julien Hamonier, laurent Loosveldt

TL;DR

The paper addresses the lack of practical methods to numerically simulate generalized Hermite processes of order $d\ge 2$, including non-Gaussian cases like the Rosenblatt process. It develops a wavelet-type random series representation and constructs a concrete simulation sequence $S^{(d)}_{\mathbf{h},J}(t)$ that converges almost surely to $X^{(d)}_{\mathbf{h}}(t)$ on compact intervals with rate $2^{-J(h_1+\cdots+h_d-d+1/2)}$, up to polynomial factors in $J$. A continuous-path interpolant $\widetilde{S}^{(d)}_{\mathbf{h},J}$ is shown to achieve the same convergence rate under a technical condition on the scale parameter $a$, and explicit algorithms are provided for the Rosenblatt process, Hermite order-3, and generalized Hermite order-3, with simulations illustrating the method. The work enables practical testing and estimation of Hurst parameters in higher-order Wiener chaos models and broadens numerical tools for non-Gaussian self-similar processes.

Abstract

Hermite processes are paradigmatic examples of stochastic processes which can belong to any Wiener chaos of an arbitrary order; the wellknown fractional Brownian motion belonging to the Gaussian first order Wiener chaos and the Rosenblatt process belonging to the non-Gaussian second order Wiener chaos are two particular cases of them. Except these two particular cases no simulation method for sample paths of Hermite processes is available so far. The goal of our article is to introduce a new method which potentially allows to simulate sample paths of any Hermite process and even those of any generalized Hermite process. Our starting point is the representation for the latter process as random wavelet-typeseries, obtained in our very recent paper [3]. We construct from it a "concrete" sequence of piecewise linear continuous random functions which almost surely approximate sample paths of this process for the uniform norm on any compact interval, and we provide an almost sure estimate of the approximation error. Then, for the Rosenblatt process and more importantly for the third order Hermite process, we propose algorithms allowing to implement this sequence and we illustrate them by several simulations. Python routines implementing these synthesis procedures are available upon request.

Numerical simulation of Generalized Hermite Processes

TL;DR

The paper addresses the lack of practical methods to numerically simulate generalized Hermite processes of order , including non-Gaussian cases like the Rosenblatt process. It develops a wavelet-type random series representation and constructs a concrete simulation sequence that converges almost surely to on compact intervals with rate , up to polynomial factors in . A continuous-path interpolant is shown to achieve the same convergence rate under a technical condition on the scale parameter , and explicit algorithms are provided for the Rosenblatt process, Hermite order-3, and generalized Hermite order-3, with simulations illustrating the method. The work enables practical testing and estimation of Hurst parameters in higher-order Wiener chaos models and broadens numerical tools for non-Gaussian self-similar processes.

Abstract

Hermite processes are paradigmatic examples of stochastic processes which can belong to any Wiener chaos of an arbitrary order; the wellknown fractional Brownian motion belonging to the Gaussian first order Wiener chaos and the Rosenblatt process belonging to the non-Gaussian second order Wiener chaos are two particular cases of them. Except these two particular cases no simulation method for sample paths of Hermite processes is available so far. The goal of our article is to introduce a new method which potentially allows to simulate sample paths of any Hermite process and even those of any generalized Hermite process. Our starting point is the representation for the latter process as random wavelet-typeseries, obtained in our very recent paper [3]. We construct from it a "concrete" sequence of piecewise linear continuous random functions which almost surely approximate sample paths of this process for the uniform norm on any compact interval, and we provide an almost sure estimate of the approximation error. Then, for the Rosenblatt process and more importantly for the third order Hermite process, we propose algorithms allowing to implement this sequence and we illustrate them by several simulations. Python routines implementing these synthesis procedures are available upon request.

Paper Structure

This paper contains 9 sections, 7 theorems, 79 equations, 7 figures.

Table of Contents

  1. Introduction and motivation
  2. Background and statement of the main results
  3. Proofs of the main results
  4. Proof of Theorem \ref{['thm:main']}
  5. Proof of Theorem \ref{['thm:main2']}
  6. Examples
  7. Simulation of the Rosenblatt process
  8. Simulation of the Hermite process of order 3
  9. Simulation of the generalized Hermite process of order 3

Key Result

Theorem 2.6

The random series is almost surely, for each $J \in \mathbb{N}$, uniformly convergent in $t$ on each compact interval $I \subset \mathbb{R}_+$. Moreover, for all such $I$, there exists an almost surely finite random variable (depending on $I$) for which one has, almost surely, for each $J \in \mathbb{N}$,

Figures (7)

  • Figure 1: Sample paths of the Rosenblatt process of Hurst parameter $0,6$ (blue), $0,7$ (orange), $0,8$ (green) and $0,9$ (red).
  • Figure 2: Sample paths of the Hermite process of order $3$ of Hurst parameter $0,6$ (blue), $0,7$ (orange), $0,8$ (green) and $0,9$ (red).
  • Figure 3: Sample paths of the Fractional Brownian Motion (blue), the Rosenblatt process (orange) and the Hermite process of order $3$ (green) with Hurst parameter $0,6$.
  • Figure 4: Sample paths of the Fractional Brownian Motion (blue), the Rosenblatt process (orange) and the Hermite process of order $3$ (green) with Hurst parameter $0,7$.
  • Figure 5: Sample paths of the Fractional Brownian Motion (blue), the Rosenblatt process (orange) and the Hermite process of order $3$ (green) with Hurst parameter $0,8$.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 14 more