Stable Distribution

In this paper we investigate ad hoc networks based on impulse radio ultra wideband up converted in the 60 GHz band. It is a extension of the paper presented in the 2nd sympo- sium autonomous and spontaneous networks of the TELECOM institute. Due to multiple access, the interference distribution is not Gaussian. Considering the predominance of rare events in the

A theory which describes the share price evolution at financial markets as a continuoustime random walk Scalas (2000); Mainardi (2000); Sabatelli (2002); Raberto (2002) has been generalized in order to take into account the dependence of waiting times t on price returns x. A joint probability density function (pdf) φ X,T (x, t) which uses the concept of a Lévy stable distribution is worked out. The theory is fitted to high-frequency US $/Japanese Yen exchange rate and low-frequency 19th century Irish stock data. The theory has been fitted both to price return and to waiting time data and the adherence to data, in terms of the χ 2 test statistic, has been improved when compared to the old theory Sabatelli (2002).

Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide probability density functions, evolving on time or variable in space, which are related to the peculiar class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions are known to play a key role.

This paper examines the portfolio optimization of energy futures by using the STARR ratio that can evaluate the risk and return relationship for skewed distributed returns. We model the price returns for energy futures by using the ARMA(1,1)-GARCH(1,1)-PCA model with stable distributed innovations that reflects the characteristics of energy: mean reversion, heteroskedasticity, seasonality, and spikes. Then, we propose the method for selecting the portfolio of energy futures by maximizing the STARR ratio, what we call "Winner portfolio". The empirical studies by using energy futures of WTI crude oil, heating oil, and natural gas traded on the NYMEX compare the price return models with stable distributed innovations to those with normal ones.

We consider a new family of $\R^d$-valued L\'{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.

Since the work of Mandelbrot in the 1960's there has accumulated a great deal of empirical evidence for heavy tailed models in finance. In these models, the probability of a large fluctuation falls off like a power law. The generalized central limit theorem shows that these heavy-tailed fluctuations accumulate to a stable probability distribution. If the tails are not too heavy then the variance is finite and we find the familiar normal limit, a special case of stable distributions. Otherwise the limit is a nonnormal stable distribution, whose bell-shaped density may be skewed, and whose probability tails fall off like a power law. The most important model parameter for such distributions is the tail thickness α, which governs the rate at which the probability of large fluctuations diminishes. A smaller value of α means that the probability tails are fatter, implying more volatility. In fact, when α < 2 the theoretical variance is infinite. A portfolio can be modeled using random vectors, where each entry of the vector represents a different asset. The tail parameter α usually depends on the coordinate. The wrong coordinate system can mask variations in α, since the heaviest tail tends to dominate. A judicious choice of coordinate system is given by the eigenvectors of the sample covariance matrix. This isolates the heaviest tails, associated with the largest eigenvalues, and allows a more faithful representation of the dependence between assets.

After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained with different approaches.

This article deals with the estimation of the parameters of an α-stable distribution by the indirect inference method with the skewed-t distribution as an auxiliary model. The latter distribution appears as a good candidate for an auxiliary model since it has the same number of parameters as the α-stable distribution, with each parameter playing a similar role. To improve the properties of the estimator in finite sample, we use a variant of the method called Constrained Indirect Inference. In a Monte Carlo study, we show that this method delivers estimators with good properties in finite sample. In particular they are much more efficient than two other prevalent methods based on the characteristic function and the empirical quantiles. We provide an empirical application to hedge fund returns.

Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide probability density functions, evolving on time or variable in space, which are related to the peculiar class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions are known to play a key role.

Asymmetric Laplace distributions have received much attention in recent years. It can be used in modeling currency exchange rate, interest rate, stock price changes, etc. But no time-series models with asymmetric Laplace marginal are yet developed. Present work aims at developing autoregressive models with asymmetric Laplace marginal distribution. r

{ Fractional calculus allows one to generalize the linear (one dimensional) di usion equation by replacing either the rst time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized di usion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional di usion a random-walk model is also proposed.

We introduce a practical alternative to Gaussian risk factor distributions based on Svetlozar Rachev's work on Stable Paretian Models in Finance (see and called the Stable Distribution Framework. In contrast to normal distributions, stable distributions capture the fat tails and the asymmetries of real-world risk factor distributions. In addition, we make use of copulas, a generalization of overly restrictive linear correlation models, to account for the dependencies between risk factors during extreme events, and multivariate ARCH-type processes with stable innovations to account for joint volatility clustering. We demonstrate that the application of these techniques results in more accurate modeling of extreme risk event probabilities, and consequently delivers more accurate risk measures for both trading and risk management. Using these superior models, VaR becomes a much more accurate measure of downside risk. More importantly Stable Expected Tail Loss (SETL) can be accurately calculated and used as a more informative risk measure for both market and credit portfolios. Along with being a superior risk measure, SETL enables an elegant approach to portfolio optimization via convex optimization that can be solved using standard scalable linear programming software. We show that SETL portfolio optimization yields superior risk adjusted returns relative to Markowitz portfolios. Finally, we introduce an alternative investment performance measurement tools: the Stable Tail Adjusted Return Ratio (STARR), which is a generalization of the Sharpe ratio in the Stable Distribution Framework.

Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.

In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(\alpha)$, has a non trivial distribution iff $p=\alpha=2$. The case for $2 > p > \alpha$ and $p \le \alpha < 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Cs\"org\"o et al. who showed Donsker's theorem for $Y_{n,2}(\cdot)$, i.e., for $p=2$, holds iff $\alpha =2$ and identified the limiting process as standard Brownian motion in sup norm.

The emergence of CDS indices and corresponding credit risk transfer markets with high liquidity and narrow bid-ask spreads has created standard benchmarks for market credit risk and correlation against which portfolio credit risk models can be calibrated. Integrated risk management for correlation dependent credit derivatives, such as single-tranches of synthetic CDOs, requires an approach that adequately reflects the joint default behavior in the underlying credit portfolios. Another important feature for such applications is a flexible model architecture that incorporates the dynamic evolution of underlying credit spreads. In this paper, we present a model that can be calibrated to quotes of CDS index-tranches in a statistically sound way and simultaneously has a dynamic architecture to provide for the joint evolution of distance-to-default measures. This is accomplished by replacing the normal distribution by smoothly truncated α-stable (STS) distributions in the Black/Cox version of the Merton approach for portfolio credit risk. This is possible due to the favorable features of this distribution family, namely, consistent application in the Black/Scholes no-arbitrage framework and the preservation of linear correlation concepts. The calibration to spreads of CDS index tranches is accomplished by a genetic algorithm. Our distribution assumption reflects the observed leptokurtic and asymmetric properties of empirical asset returns since the STS distribution family is basically constructed from α-stable distributions. These exhibit desirable statistical properties such as domains of attraction and the application of the generalized central limit theorem. Moreover, STS distributions fulfill technical restrictions like finite (exponential) moments of arbitrary order. In comparison to the performance of the basic normal distribution model which lacks tail dependence effects, our empirical analysis suggests that our extension with a heavy-tailed and highly peaked distribution provides a better fit to tranche quotes for the iTraxx IG index. Since the underlying implicit modeling of the dynamic evolution of credit spreads leads to such results, this suggests that the proposed model is appropriate to price and hedge complex transactions that are based on correlation dependence. A further application might be integrated risk management activities in debt portfolios where concentration risk is dissolved by means of portfolio credit risk transfer instruments such as synthetic CDOs.

Several approaches have been considered to model the heavy tails and asymmetric effect on stocks returns volatility. The most commonly used models are the Exponential Generalized AutoRegressive Conditional Heteroskedasticity (EGARCH), the Threshold GARCH (TGARCH), and the Asymmetric Power ARCH (APARCH) which, in their original form, assume a Gaussian distribution for the innovations. In this paper we propose the estimation of all these asymmetric models on empirical distributions of the Standard & Poor's (S&P) 500 and the Financial Times Stock Exchange (FTSE) 100 daily returns, assuming the Student's t and the stable Paretian (with α < 2) distributions for innovations. To the authors' best knowledge, analysis of the EGARCH and TGARCH assuming innovations with α-stable distribution have not yet been reported in the literature. The results suggest that this kind of distributions clearly outperforms the Gaussian case. However, when α-stable and Student's t distributions are compared, a general conclusion should be avoided as the goodness-of-fit measures favor the αstable distribution in the case of S&P 500 returns and the Student's t distribution in the case of FTSE 100.

This paper introduces a new multiscale speckle reduction method based on the extraction of wavelet interscale dependencies to visually enhance the medical ultrasound images and improve clinical diagnosis. The logarithm of the image is first transformed to the oriented dual-tree complex wavelet domain. It is then shown that the adjacent subband coefficients of the log-transformed ultrasound image can be successfully modeled using the general form of bivariate isotropic stable distributions, while the speckle coefficients can be approximated using a zero-mean bivariate Gaussian model. Using these statistical models, we design a new discrete bivariate Bayesian estimator based on minimizing the mean square error (MSE). To assess the performance of the proposed method, four image quality metrics, namely signal-to-noise ratio, MSE, coefficient of correlation, and edge preservation index, were computed on 80 medical ultrasound images. Moreover, a visual evaluation was carried out by two medical experts. The numerical results indicated that the new method outperforms the standard spatial despeckling filters, homomorphic Wiener filter, and new multiscale

We analyze cross-household inflation dispersion in Europe using "fictitious" monthly inflation rates for several household categories (grouped according to income levels, household size, socio-economic status, age) for the period from 1997 to 2008. Our analysis is carried out on a panel of 23 up to 27 household-specific inflation rates per country for 15 countries. In the first part of the paper, we employ time series and related non-stationary panel approaches to shed light on cross-country differences in inflation inequality with respect to the number of driving forces in the panel. In particular, we focus on the degree of persistence of the household-specific inflation rates and their the adjustment behaviour towards the inflation rate of a "representative household". In the second part of the paper, we pool over the full sample of all countries and test if and by how much certain household categories across Europe are more prone to significant inflation differentials and significant differences in the volatility of inflation. Furthermore we search for the presence of clusters with respect to inflation susceptibility. On the national level, we find evidence for the existence of one main driving factor driving the non-stationarity of the panel and evidence for a single co-integration vector. Persistence of deviations, however, is high, and the adjustment speed towards the "representative household" is low. Even if there is no concern about a long-run stable distribution, at least in the short-to medium run deviations tend to last. On the European level, we find small but significant differences (mainly along income levels), we can separate 5 clusters and two main driving forces for the differences in the overall panel. All in all, even if differences are relatively small, they are not negligible and persistent enough to represent a serious matter of debate for economic and social policy. The positions do not necessarily reflect those of other persons in the institutions the authors might be affiliated with. Thanks to Ingrid Grössl and seminar participants at DG-ECFIN for helpful comments. Thanks to Daniel Triet, Artur Tarassow and Phillip Poppitz for outstanding research assistance. All remaining errors are ours.

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the "KR distribution". We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr-Geman-Madan-Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the "change of measure" for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.

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We develop and test models for the population dynamics of species that undergo regular alternations of generations between independent, free-living, haploid and diploid phases. The models are patterned after the dioecious, haploid-diploid lifecycle of many marine algae. If the two phases have equal demographic rates, all models (with or without density dependence) predict a stable distribution with a ratio of :1 (ϳ60% ͙2 haploids and 40% diploids). We find that observable deviations from this distribution can occur when the demographic rates (mortality, fecundity) between phases vary widely. Field surveys of three macroalgal species with independent, free-living haploids and diploids, Mazzaella flaccida, M. laminarioides, and M. splendens, reveal a consistent pattern of haploid dominance that significantly exceeds this null model expectation (72%, 76%, and 66%, respectively). These species also exhibit significant variation in haploid frequency among sites. We estimated the per capita fecundity and mortality rates for each phase of both M. flaccida and M. laminarioides and used this information to address two basic questions: (1) Which parameter better explains the overall high relative haploid abundance for each species? and (2) Which parameter better explains the variation in relative haploid abundance among sites?

In this paper, we discuss the issue of estimation of the parameters of stable laws. We present an overview of the known methods and compare them on samples of different sizes and for different values of the parameters. Performance tables are provided.

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assuming infinitely divisible distributions, we

Several studies that have investigated a few stocks have found that the spacing between consecutive financial transactions (referred to as trade duration) tend to exhibit long-range dependence, heavy tailedness, and clustering. In this study, we empirically investigate whether a larger sample of stocks exhibit those characteristics. Comparing goodness of fit in modeling trade duration data for stable distribution and fractional stable noise based on a procedure applying bootstrap methods developed by the authors, the empirical results in this paper suggest that the fractional stable noise and stable distribution dominate alternative distributional assumptions (i.e., lognormal distribution, fractional Gaussian noise, exponential distribution, and Weibull distribution) in our study.

Opportunistic spectrum access creates the opening of under-utilized portions of the licensed spectrum for reuse, provided that the transmissions of secondary radios do not cause harmful interference to primary users. Such a system would require secondary users to be cognitive-they must accurately detect and rapidly react to varying spectrum usage. Therefore, it is important to characterize the effect of cognitive network interference due to such secondary spectrum reuse. In this paper, we propose a new statistical model for aggregate interference of a cognitive network, which accounts for the sensing procedure, secondary spatial reuse protocol, and environment-dependent conditions such as path loss, shadowing, and channel fading. We first derive the characteristic function and cumulants of the cognitive network interference at a primary user. Using the theory of truncated-stable distributions, we then develop the statistical model for the cognitive network interference. We further extend this model to include the effect of power control and demonstrate the use of our model in evaluating the system performance of cognitive networks. Numerical results show the effectiveness of our model for capturing the statistical behavior of the cognitive network interference. This work provides essential understanding of interference for successful deployment of future cognitive networks.

Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. In recent empirical studies of stock market indices it was examined whether the distribution P (r) of returns r(τ) after some time τ can be described by a (truncated) Lévy-stable distribution L α (r) with some index 0 < α ≤ 2. While the Lévy distribution cannot be expressed in a closed form, one can identify its parameters by testing the dependence of the central peak height on τ as well as the power-law decay of the tails. In an earlier study [Mantegna and Stanley, Nature 376, 46 (1995)] it was found that the behavior of the central peak of P (r) for the Standard & Poor 500 index is consistent with the Lévy distribution with α = 1.4. In a more recent study [Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999)] it was found that the tails of P (r) exhibit a power-law decay with an exponent α ∼ = 3, thus deviating from the Lévy distribution. In this paper we study the distribution of returns in a generic model that describes the dynamics of stock market indices. For the distributions P (r) generated by this model, we observe that the scaling of the central peak is consistent with a Lévy distribution while the tails exhibit a power-law distribution with an exponent α > 2, namely beyond the range of Lévy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results.

The α-stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its many useful properties, including a central limit theorem, are especially appreciated in the financial field. However, estimation difficulties have up to now hindered its diffusion among practitioners. In this paper we propose an indirect estimation approach to stochastic volatility models with α-stable innovations that exploits, as auxiliary model, a GARCH(1,1) with t-distributed innovations. We consider both cases of heavytailed noise in the returns or in the volatility. The approach is illustrated by means of a detailed simulation study and an application to currency crises.

In this paper we compare different approaches to compute VaR for heavy tailed return series. Using data from the Italian market, we show that almost all the return series present statistically significant skewness and kurtosis. We implement (i) the stable models proposed by , (ii) an alternative to the Gaussian distributions based on a Generalized Error Distribution , (iii) a nonparametric model proposed by . All the model are then submitted to backtest on out-of-sample data in order to assess their forecasting power. We observe that when the percentiles are low, all the models tested produce results that are dominant compared to the standard RiskMetrics model.

Large variations in stock prices happen with sufficient frequency to raise doubts about existing models, which all fail to account for non-Gaussian statistics. We construct simple models of a stock market, and argue that the large variations may be due to a crowd effect, where agents imitate each other&#39;s behavior. The variations over different time scales can be related to

The integration of quantitative asset allocation models and the judgment of portfolio managers and analysts (i.e., qualitative view) dates back to papers by Black and Litterman [4], [5], [6]. In this paper we improve the classical Black-Litterman model by applying more realistic models for asset returns (the normal, the t-student, and the stable distributions) and by using alternative risk measures (dispersion-based risk measures, value at risk, conditional value at risk). Results are reported for monthly data and goodness of the models are tested through a rolling window of fixed size along a fixed horizon. Finally, we find that incorporation of the views of investors into the model provides information as to how the different distributional hypotheses can impact the optimal composition of the portfolio.

α-stable distributions are utilised as models for heavy-tailed noise in many areas of statistics, finance and signal processing engineering. However, in general, neither univariate nor multivariate α-stable models admit closed form densities which can be evaluated pointwise. This complicates the inferential procedure. As a result, α-stable models are practically limited to the univariate setting under the Bayesian paradigm, and to bivariate models under the classical framework. In this article we develop a novel Bayesian approach to modelling univariate and multivariate α-stable distributions based on recent advances in "likelihood-free" inference. We present an evaluation of the performance of this procedure in 1, 2 and 3 dimensions, and provide an analysis of real daily currency exchange rate data. The proposed approach provides a feasible inferential methodology at a moderate computational cost.

This paper discusses two optimal allocation problems. We consider different hypotheses of portfolio selection with stable distributed returns for each of them. In particular, we study the optimal allocation between a riskless return and risky stable distributed returns. Furthermore, we examine and compare the optimal allocation obtained with the Gaussian and the stable non-Gaussian distributional assumption for the risky return. KEY WORDS: optimal allocation, stochastic dominance, risk aversion, measure of risk, a stable distribution, domain of attraction, sub-Gaussian stable distributed, fund separation, normal distribution, mean variance analysis, safety-first analysis.

We study the daily return distributions for 22 industry stock indexes on the Tai-wan Stock Exchange under the unconditional homoskedastic independent, identically distributed and the conditional heteroskedastic GARCH models. Two distribution hypotheses are tested: the Gaussian and the stable Paretian distributions. The performance of the stable Paretian distribution is better than that of the Gaussian distribution. A back-testing example is provided to give evidence on the superiority of the stable ARMA-GARCH to the normal ARMA-GARCH.

A new variant of Local Linearization (LL) method is proposed for the numerical (strong) solution of differential equations driven by (additive) alpha-stable Lévy motions. This is studied through simulations making emphasis in comparison with the Euler method from the viewpoint of numerical stability. In particular, a number of examples of stiff equations are shown in which the Euler method has explosive behavior while the LL method correctly reproduces the dynamics of the exact trajectories.

This paper proposes several parametric models to compute the portfolio VaR and CVaR in a given temporal horizon and for a given level of confidence. Firstly, we describe extension of the EWMA RiskMetrics model considering conditional elliptically distributed returns. Secondly, we examine several new models based on different stable Paretian distributional hypotheses of return portfolios. Finally, we discuss the applicability

To understand the evolution of diverse species, theoretical studies using a Lotka-Volterra type direct competition model had shown that concentrated distributions of species in continuous trait space often occurs. However, a more mechanistic approach is preferred because the competitive interaction of species usually occurs not directly but through competition for resource. We consider a chemostat-type model where species consume resource that are constantly supplied. Continuous traits in both consumer species and resource are incorporated. Consumers utilize resource whose trait values are similar with their own. We show that, even when resource-supply has a continuous distribution in trait space, a positive continous distribution of consumer trait is impossible. Self-organized generation of distinct species occurs. We also prove global convergence to the evolutionarily stable distribution.

This paper is devoted to the analysis of some fundamental problems of linear elasticity in 1D continua with self-similar interparticle interactions. We introduce a self-similar continuous field approach where the self-similarity is reflected by equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit $\delta$-force. In the dynamic framework we derive the solution of the {\it Cauchy problem} and the retarded Green's function. We deduce the distribution of a self-similar variant of diffusion problem with L\'evi-stable distributions as solutions with infinite mean fluctuations describing the statistics L\'evi-flights. We deduce a hierarchy of solutions for the self-similar Poisson's equation which we call "self-similar potentials". These non-local singular potentials are in a sense self-similar analogues to the 1D-Dirac's $\delta$-function. The approach can be the starting point to tackle a variety of scale invariant interdisciplinary problems.

This paper examines the portfolio optimization of energy futures by using the STARR ratio that can evaluate the risk and return relationship for skewed distributed returns. We model the price returns for energy futures by using the ARMA(1,1)-GARCH(1,1)-PCA model with stable distributed innovations that reflects the characteristics of energy: mean reversion, heteroskedasticity, seasonality, and spikes. Then, we propose the method for selecting the portfolio of energy futures by maximizing the STARR ratio, what we call "Winner portfolio". The empirical studies by using energy futures of WTI crude oil, heating oil, and natural gas traded on the NYMEX compare the price return models with stable distributed innovations to those with normal ones.

This study develops a multi-speed numerical approach to simulate the two-dimensional superdiffusion described by the fractional-derivative equation, by extending the standard Lattice-Boltzmann method. To approximate the large displacements of particles due to super-diffusion, the velocity components assigned to each particle need to be refined. This is done by modifying the equilibrium distribution function of Lattice-Boltzmann method. The convergence and accuracy analysis are supplied by Chapman-Enskog expansion. Numerical examples show the applicability of the novel multi-speed model for approximating the anomalous super-diffusion.

Keywords: supper diffusion, Lattice-Boltzmann method, fractional differential equation, stable distribution

A theory which describes the share price evolution at financial markets as a continuoustime random walk Scalas (2000); Mainardi (2000); Sabatelli (2002); Raberto (2002) has been generalized in order to take into account the dependence of waiting times t on price returns x. A joint probability density function (pdf) φ X,T (x, t) which uses the concept of a Lévy stable distribution is worked out. The theory is fitted to high-frequency US $/Japanese Yen exchange rate and low-frequency 19th century Irish stock data. The theory has been fitted both to price return and to waiting time data and the adherence to data, in terms of the χ 2 test statistic, has been improved when compared to the old theory Sabatelli (2002).

Linear filtering theory has been largely motivated by the characteristics of Gaussian signals. In the same manner, the proposed myriad filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non-Gaussian impulsive environments that can appear in practice. We introduce several important properties of the myriad filter and prove its optimality in the family of -stable distributions.

The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in time. In these cases the related stochastic processes are self-similar and are simply referred to as fractional diffusion processes. We provide some integral formulas involving the distributions of these processes that can be interpreted in terms of subordination laws.