Tempered fractional multistable motion and tempered multifractional stable motion

This work defines two classes of processes, that we term tempered fractional multistable motion and tempered multifractional stable motion. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These processes may provide useful models for data that exhibit both dependence and varying local regularity/intensity of jumps.

AN ENTROPY BASED IN WAVELET LEADERS TO QUANTIFY THE LOCAL REGULARITY OF A SIGNAL AND ITS APPLICATION TO ANALIZE THE DOW JONES INDEX

Local regularity analysis is useful in many fields, such as financial analysis, fluid mechanics, PDE theory, signal and image processing. Different quantifiers have been proposed to measure the local regularity of a function. In this paper we present a new quantifier of local regularity of a signal: the pointwise wavelet leaders entropy. We define this new measure of regularity by combining the concept of entropy, coming from the information theory and statistical mechanics, with the wavelet leaders coefficients. Also we establish its inverse relation with one of the well-known regularity exponents, the pointwise Hölder exponent. Finally, we apply this methodology to the financial data series of the Dow Jones Industrial Average Index, registered in the period 1928–2011, in order to compare the temporal evolution of the pointwise Hölder exponent and the pointwise wavelet leaders entropy. The analysis reveals that temporal variation of these quantifiers reflects the evolution of the Dow Jones Industrial Average Index and identifies historical crisis events. We propose a new approach to analyze the local regularity variation of a signal and we apply this procedure to a financial data series, attempting to make a contribution to understand the dynamics of financial markets.