[Paper Review] A class of spatio-temporal and causal stochastic processes, with application to multiscaling and multifractality

Jürgen Schmiegel, Ole E. Barndorff–Nielsen+1 more|ArXiv.org|Dec 13, 2005

Stochastic processes and financial applications14 references17 citations

TL;DR

This paper introduces a general class of causal, spatio-temporal stochastic processes based on independently scattered Lévy random measures, enabling exact analytical computation of n-point correlations. It establishes that a multiscaling process derived from power-law two-point correlations exhibits multifractal scaling, with integral moments scaling as $ l^{-\mu(n)} $ in the large-scale limit, confirming its multifractal nature through rigorous asymptotic analysis and error bounds.

ABSTRACT

We present a general class of spatio-temporal stochastic processes describing the causal evolution of a positive-valued field in space and time. The field construction is based on independently scattered random measures of Levy type whose weighted amplitudes are integrated within a causality cone. General n-point correlations are derived in closed form. As a special case of the general framework, we consider a causal multiscaling process in space and time in more detail. The latter is derived from, and completely specified by, power-law two-point correlations, and gives rise to scaling behaviour of both purely temporal and spatial higher-order correlations. We further establish the connection to classical multifractality and prove the multifractal nature of the coarse-grained field amplitude.

Motivation & Objective

• To develop a unified, causal framework for modeling spatio-temporal stochastic processes with prescribed correlation structures.
• To extend existing multifractal and multiscaling models to continuous, causal, and spatially extended processes.
• To analytically derive and validate the multifractal scaling behavior of coarse-grained field amplitudes in such processes.
• To establish a connection between power-law two-point correlations and higher-order scaling via generalized fusion rules.
• To provide a rigorous error analysis for approximations of n-point correlation integrals in the large-scale limit.

Proposed method

• Construction of the process via integration of weighted amplitudes of a Lévy basis over a causality cone (ambit set), ensuring temporal and spatial causality.
• Use of independently scattered random measures of Lévy type to ensure generalizable n-point correlation structure independent of specific realization.
• Derivation of closed-form expressions for n-point correlations using recursive integration over ordered time/space intervals.
• Application of generalized fusion rules to express n-point correlation functions as products of power-law terms in inter-event distances.
• Asymptotic analysis of integral moments $ M_n^{(s)}(x,l) $, showing $ \tilde{M}_n(l) \propto l^{-\mu(n)} $ in the large-scale limit.
• Error estimation via bounding neglected integrals with distances $ < l_{\text{scal}} $, proving relative error vanishes as $ l \to \infty $ under $ n > \mu(n) $.

Experimental results

Research questions

• RQ1 Can a general class of causal, spatio-temporal stochastic processes be constructed with analytically tractable n-point correlations?
• RQ2 How can multiscaling behavior in both temporal and spatial correlations be derived from power-law two-point correlations?
• RQ3 What conditions ensure that a coarse-grained field amplitude exhibits multifractal scaling in such a process?
• RQ4 How accurate is the large-scale approximation of n-point correlation integrals, and what is the asymptotic behavior of the relative error?
• RQ5 Can the multifractal nature of the process be rigorously proven using the derived correlation structure and scaling exponents?

Key findings

• The n-point correlation function is derived in closed form using recursive integration over ordered time/space intervals, with the structure determined by generalized fusion rules.
• The integral moment $ \tilde{M}_n(l) $ scales asymptotically as $ l^{-\mu(n)} $ in the large-scale limit, confirming multifractal scaling behavior.
• The relative error between the exact and approximate n-point correlation integrals vanishes as $ l \to \infty $, provided $ n > \mu(n) $, which holds due to monotonicity of $ l^n M_n^{(s)}(x,l) $.
• The scaling exponent $ \mu(n) $ is expressed as $ \mu(n) = \tau(2) \frac{\mathrm{K}[n] - n\mathrm{K}[1]}{\mathrm{K}[2] - 2\mathrm{K}[1]} $, linking it to the structure of the underlying Lévy basis.
• The multifractal nature of the coarse-grained field amplitude is proven by showing that $ \tilde{M}_n(l) \propto l^{-\mu(n)} $ with $ \mu(n) $ non-linear in $ n $, a hallmark of multifractality.
• The results are independent of the small-scale statistics as long as they are finite, ensuring robustness of the large-scale scaling behavior.