Local Holder Regularity Research Articles

Do Sentiment and Exchange Rate Share Memories? An Application of Multifractional Process Modeling

We use the notion of local Holder regularity to investigate the roughness of the pattern of EUR/USD exchange rate process as well as the corresponding investor sentiment dynamic process. Specifically, we use the pointwise Holder exponent to measure the local Holder regularity of a multifractional process. This application extends the increment ratio method to estimate the pointwise Holder exponent of multifractional Brownian motion function. Our testing and comparison results show that both the exchange rate and the sentiment proxy's local Holder regularities are time-varying and are more homogeneous during crises periods. More importantly, in contrast to the optimistic sentiment, the roughness of the pessimistic sentiment dynamic process displays only slightly higher similarity to that of EUR/USD exchange rate. These results suggest that the EUR/USD exchange rate process and the pessimistic sentiment dynamic process share a common pattern of smoothness, which implies that they share a similar memory pattern.

Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence.

An algorithm is described that can generate random variants of a time series while preserving the probability distribution of original values and the pointwise Hölder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the iterated amplitude adjusted wavelet transform can be used to generate bootstrapped versions of multifractal data, and because it preserves the pointwise Hölder regularity but not the local Hölder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes.

An estimate for the Hölder constant for weak solutions to m-Hessian equations in a closed domain

The smoothness properties of weak solutions to the Dirichlet problem for m-Hessian equations are studied. Namely, fully nonlinear second-order equations of the form $$ tr_m u_{xx} = f^m $$ in a domain Ω ⊂ Rn are considered; here, trmuxx is the sum of all principal minors of the matrix uxx for 1 ⩽ m ⩽ n. Such equations were considered in the 1980s by N.M. Ivochkina, L. Nirenberg, L. Caffarelly, J. Spruck, and N.V. Krylov for f ∈ C4(\( \bar \Omega \)), f > 0. In 1997, N. Trudinger introduced the notion of an approximative solution to an m-Hessian equation and outlined the proof of its local Holder regularity for f belonging to one of the Lebesgue spaces. The behavior of an approximative solution to the Dirichlet problem near the boundary provided that f ∈ Ln(Ω), f ⩾ 0, and f ∈ Lp for p ⩾ n near the boundary is analyzed. It is shown that the solution approaches the boundary value at rate dα, where d is the distance to the boundary and 0 < α < 1. The dependence of α on p is also described. Moreover, the simplest proof of the Holder regularity of a weak solution in a closed domain is given. Methods developed by O.A. Ladyzhenskaya and N.N. Ural’tseva and Trudinger’s approach are used.

Smoothness-constrained quantization for wavelet image compression

Uniform quantization of wavelet coefficients introduces perceptually disturbing artifacts by decreasing the visual smoothness. In this paper, the local Holder regularity is used to quantify these changes in visual smoothness. A modified uniform quantization scheme that constrains local regularity reduction is proposed. The quantizer modification is applicable to any uniform quantization scheme and requires no additional side information to be transmitted to the decoder, unlike standard deadzone quantizers. The reconstructed images show a noticeable perceptual improvement as well as an increased PSNR relative to uniform quantization.

Besov spaces and the multifractal hypothesis

Parisi and Frisch proposed some time ago an explanation for “multiscaling” of turbulent velocity structure functions in terms of a “multifractal hypothesis,” i.e., they conjecture that the velocity field has local Holder exponents in a range [hmin,hmax], with exponents <h occurring on a setS(h) with a fractal dimensionD(h). Heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the Legendre transform of the codimensiond-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field hasL p -mean Holder indexs, i.e., if it lies in the Besov spaceB p s,∞ , then local Holder regularity is satisfied. Ifs<d/p, then the hypothesis is true in a generalized sense of Holder space with negative exponents and we discuss the proper definition of local Holder classes of negative index. Finally, if a certain “box-counting dimension” exists, then the Legendre transform of its codimension gives the scaling exponentz p , and, more generally, the maximal Besov index of order,p, ass p =z p /p. Our method of proof is derived from a recent paper of S. Jaffard using compactly-supported, orthonormal wavelet bases and gives an extension of his results. We discuss implications of the theorems for ensemble-average scaling and fluid turbulence.