[Paper Review] Gaussian processes, bridges and membranes extracted from selfsimilar random fields

Maik Görgens, Ingemar Kaj|arXiv (Cornell University)|Oct 2, 2014

Financial Risk and Volatility Modeling16 references3 citations

TL;DR

This paper introduces a novel construction of self-similar Gaussian random fields using shot-noise mechanisms driven by Gaussian white noise or random balls white noise, enabling the explicit extraction of fractional Brownian motion, bridges, and membranes on bounded domains. The key contribution is a unified framework that generates Gaussian processes with prescribed self-similarity and boundary behavior via measure-based indexing and Riesz transform extensions.

ABSTRACT

We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on $\mathbb{R}^d$ and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on $\mathbb{R}^d$, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on $\mathbb{R}^d$. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.

Motivation & Objective

• To develop a general method for constructing self-similar Gaussian generalized random fields on R^d using shot-noise mechanisms.
• To extend the range of self-similarity indices H beyond previous limits using Riesz transforms and signed measures.
• To extract pointwise-defined Gaussian processes such as fractional Brownian motion, bridges, and membranes from these fields.
• To provide a new representation of fractional Brownian motion using random balls white noise.
• To construct Gaussian membranes that vanish on the boundary of a bounded domain D via hard-thinning of random balls.

Proposed method

• Construct self-similar Gaussian fields via shot-noise mechanisms driven by Gaussian white noise M_d or random balls white noise W_β on R^d × R_+.
• Use signed measures μ_t indexed by t ∈ R^d to extract pointwise processes Y_t = X(μ_t) from the underlying field X.
• Apply the Riesz transform (−Δ)^{-m/2} to extend the range of self-similarity indices H to any H ∉ Z (d ≥ 2) or H ∉ ½Z (d = 1).
• Define membranes as processes X_t that vanish in probability as t approaches the boundary ∂D of a bounded domain D.
• Implement hard-thinning of random balls by discarding balls not fully contained within D, yielding boundary-vanishing membranes.
• Use covariance analysis and limit arguments involving τ(ε)^2 to derive exact forms of covariance functions for the extracted processes.

Experimental results

Research questions

• RQ1 Can self-similar Gaussian fields with H > 0 be constructed from generalized random fields using shot-noise mechanisms?
• RQ2 How can fractional Brownian motion on R^d be extracted from self-similar random fields via measure-based indexing?
• RQ3 What is the structure of Gaussian bridges and membranes that vanish on the boundary of a bounded domain D?
• RQ4 How does the random balls model with hard-thinning produce a boundary-vanishing membrane process?
• RQ5 Can the hard-thinning bridge on [0,T] be related to fractional Brownian motion via a martingale decomposition?

Key findings

• The covariance of the extracted process T'(s)T'(t) is shown to be c(|s|^{2H} + |t|^{2H} - |s-t|^{2H}) for some constant c, confirming self-similarity with index H.
• For d=1 and D=(0,T), the hard-thinning bridge has covariance E[W_β(s)W_β(t)] = f_β(s∨t) + f_β(T−s∧t) − f_β(|s−t|) − f_β(T), with f_β(x) = 2^β/(β(1−β)) x^{1−β} for 0<β<1.
• When β = −1, the hard-thinning bridge reduces to the classical Brownian bridge with covariance ½(s∧t)(T−s∨t).
• For 0<β<1, the sum W_β + Y_β, where Y_β is a stochastic integral with volatility √(x^{-β} + (T−x)^{-β}), yields a fractional Brownian motion with Hurst index H = (1−β)/2.
• The process Y_β is well-defined for −1<β<0 and yields a complex Gaussian process with covariance proportional to C_β(s,t) = s^{1−β} + t^{1−β} − |s−t|^{1−β}.
• The construction via Riesz transforms allows extension of the self-similarity index H to any value not in Z (d≥2) or ½Z (d=1), significantly broadening the scope of existing models.