Superpositions of Ornstein-Uhlenbeck type (supOU) processes form a rich class of stationary processes with a flexible dependence structure. The asymptotic behavior of the integrated and partial sum supOU processes can be, however, unusual. Their cumulants and moments turn out to have an unexpected rate of growth. We identify the property of fast growth of moments or cumulants as intermittency.
The phenomenon of intermittency has been widely discussed in physics literature. This paper provides a model of intermittency based on Lévy driven Ornstein-Uhlenbeck (OU) type processes. Discrete superpositions of these processes can be constructed to incorporate non-Gaussian marginal distributions and long or short range dependence. While the partial sums of finite superpositions of OU type processes obey the central limit theorem, we show that the partial sums of a large class of infinite long range dependent superpositions are intermittent. We discuss the property of intermittency and behavior of the cumulants for the superpositions of OU type processes.
First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator.In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" W (3) (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function Z (4) (Avram, Kyprianou and Pistorius 2004).Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "W, Z alphabet" for a great variety of first passage problems.We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem, and whose optimization may be useful for the valuation of financial companies.One interesting use of the kit is for recognizing relationships between apparently unrelated problems -see Lemma 3. Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known (which is not altogether trivial given that several related strands of literature need to be checked).Last but not least, it turned out recently that once the classic W, Z are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) Lévy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative Lévy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017), or with Omega killing (Li and Palmowski 2017.This suggests that processes combining two or three of these features could also be handled by appropriate W, Z functions.An implicit question arising from our list is to investigate the existence of similar formulas for more complicated classes of spectrally negative Markovian processes, like for example continuous branching processes with and without immigration (Kawazu and Watanabe 1971), which are characterized by two Laplace exponents. This topic deserves further investigation.
The so-called partition function is a sample moment statistic based on blocks of data and it is often used in the context of multifractal processes.It will be shown that its behaviour is strongly influenced by the tail of the distribution underlying the data either in i.i.d. and weakly dependent cases.These results will be exploited to develop graphical and estimation methods for the tail index of a distribution. The performance of the tools proposed is analyzed and compared with other methods by means of simulations and examples.
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