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In many mechanical, electrical, and general physical systems evolving over time or space, spectral analysis methods as Fast Fourier Transform (FFT), Short Term Fourier Transform (STFT), Power Spectrum Density (PSD) plays a very important role. They allow an extraction of required information content from signals in another base by decomposing it in its spectral components for further processing.In theory this approach is very powerful, even in some 'simple' or 'not too complicated' practical cases it has proven its utility and efficiency. However, for real-world applications such as mechanical modal analysis of large dimension systems including damping, noise and unpredictable excitation those signals are often so complex that it can be almost impossible to obtain a high-resolution spectral decomposition with these methods due to the time-bandwidth limitation. In this paper we describe an alternative approach for spectral analysis based on the High Order Dynamical Mode Decomposition (HODMD) and Kernel Density Spectrum (KDS). We will show that this method allows overcoming some limitations of the FFT and may be a promising approach to for a much more precisely the spectral decomposition.

Traditionally, blockchain systems involve sharing transaction information across all blockchain network participants. Clearly, this introduces barriers to the adoption of the technology by the enterprise world, where preserving the privacy of the business data is a necessity. Previous efforts to bring privacy and blockchains together either still leak partial information, are restricted in their functionality or use costly mechanisms like zk-SNARKs. In this paper, we propose the Multi-Shard Private Transaction (MSPT) protocol, a novel privacy-preserving protocol for permissioned blockchains, which relies only on simple cryptographic primitives and targeted dissemination of information to achieve atomicity and high performances.

Performance and scalability are major concerns for blockchains: permissionless systems are typically limited by slow proof of X consensus algorithms and sequential post-order transaction execution on every node of the network. By introducing a small amount of trust in their participants, permissioned blockchain systems such as Hyperledger Fabric can benefit from more efficient consensus algorithms and make use of parallel pre-order execution on a subset of network nodes. Fabric, in particular, has been shown to handle tens of thousands of transactions per second. However, this performance is only achievable for contention-free transaction workloads. If many transactions compete for a small set of hot keys in the world state, the effective throughput drops drastically. We therefore propose XOX: a novel two-pronged transaction execution approach that both minimizes invalid transactions in the Fabric blockchain and maximizes concurrent execution. Our approach additionally prevents unintentional denial of service attacks by clients re-submitting conflicting transactions. Even under fully contentious workloads, XOX can handle more than 3000 transactions per second, all of which would be discarded by regular Fabric.

Blockchain technologies are expected to make a significant impact on a variety of industries. However, one issue holding them back is their limited transaction throughput, especially compared to established solutions such as distributed database systems. In this paper, we re-architect a modern permissioned blockchain system, Hyperledger Fabric, to increase transaction throughput from 3,000 to 20,000 transactions per second. We focus on performance bottlenecks beyond the consensus mechanism, and we propose architectural changes that reduce computation and I/O overhead during transaction ordering and validation to greatly improve throughput. Notably, our optimizations are fully plug-and-play and do not require any interface changes to Hyperledger Fabric., Comment: Minor revisions based on reviewer feedback

The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler function, Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:math/0701454

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for continuously differentiable functions. Accordingly, in the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of the smooth functions that are too narrow. In this paper, we introduce a suitable definition of the Caputo derivative in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces., Comment: 20 pages

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order $\alpha ~(0 < \alpha \leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors and sample path properties are studied. Finally we introduce the $q$-Mittag-Leffler process associated with the $q$-Mittag-Leffler distribution.

We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities., Comment: 20 pages. Some text may overlap with our E-prints: arXiv:1305.3074, arXiv:1210.8414, arXiv:1104.4041

The fractional Poisson process and the Wright process (as discretization of the stable subordinator) along with their diffusion limits play eminent roles in theory and simulation of fractional diffusion processes. Here we have analyzed these two processes, concretely the corresponding counting number and Erlang processes, the latter being the processes inverse to the former. Furthermore we have obtained the diffusion limits of all these processes by well-scaled refinement of waiting times and jumps, Comment: 30 pages, 4 figures. A preliminary version of this paper was an invited talk given by R. Gorenflo at the Conference ICMS2011, held at the International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5 January 2011, devoted to Prof Mathai on the occasion of his 75 birthday

We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to continuous time random walk with subsequent passage to the diffusion limit, here we go the opposite way. Via Fourier-Laplace manipulations of the relevant fractional partial differential equation of evolution we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time from the physical time and then generating in operational time the spatial path. By inverting the generation of operational time from physical time we arrive at the method of parametric subordination. Due to the infinite divisibility of the stable subordinator, we can simulate particle paths by discretization where the generated points of a path are precise snapshots of a true path. By refining the discretization more and more fine details of a path become visible., Comment: 40 pages 14 Figures. Text overlap with our E-print arXiv:1104.4041

For the symmetric case of space-fractional diffusion processes (whose basic analytic theory has been developed in 1952 by Feller via inversion of Riesz potential operators) we present three random walk models discrete in space and time. We show that for properly scaled transition to vanishing space and time steps these models converge in distribution to the corresponding time-parameterized stable probability distribution. Finally, we analyze in detail a model, discrete in time but continuous in space, recently proposed by Chechkin and Gonchar. Concerning the inversion of the Riesz potential operator $I_0^\alpha$ let us point out that its common hyper-singular integral representation fails for $\alpha = 1. In our Section 2 we have shown that the corresponding hyper-singular representation for the inverse operator $D_0^\alpha$ can be obtained also in the critical (often excluded) case $\alpha = 1$, by analytic continuation, Comment: 32 pages

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw_1), a random walk along the line of natural time, happening in operational time; (rw_2), a random walk along the line of space, happening in operational time;(rw_3), the inversion of (rw_1), namely a random walk along the line of operational time, happening in natural time. Via the general integral equation of CTRW and appropriate rescaling, the transition to the diffusion limit is carried out for each of these three random walks. Combining the limits of (rw_1) and (rw_2) we get the method of parametric subordination for generating particle paths, whereas combination of (rw_2) and (rw_3) yields the subordination integral for the sojourn probability density in space-time fractional diffusion., Comment: 20 pages, 4 figures

We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion limit of the fractional Poisson process and as a subordinator for space-time fractional diffusion., Comment: 28 pages. Invited Lecture, Workshop on Fractional Calculus and Statistical Distributions, November 25-27, 2009, Centre for Mathematical Sciences, Pala Campus, Kerala-India. To appear in the Proceedings.

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order., Comment: 56 pages, 7 figures/eps files

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity., Comment: 40 pages, 5 Figures

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories., Comment: 33 pages, 3 Figures, 5 eps files. Second International Workshop on Analysis and Numerical Approximation of Singular Problems (IWANASP 2006), 6-8 September 2006, Aegean University in Karlovassi, Samos, Greece

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that, by a well-scaled passage to the diffusion limit, generalized diffusion processes, fractional in time as well as in space, are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure., Comment: 12 pages, 4 figures

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales., Comment: 14 pages. International Symposium on "Analytic Function Theory, Fractional Calculus and Their Applications", University of Victoria (British Columbia, Canada), 22-27 August 2005

A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies., Comment: 24 pages, 3 figures, 10 eps files

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process., Comment: 36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006; Chairmen: R. Klages, G. Radons and I.M. Sokolov

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., Comment: 46 pages, 3 figures. International Workshop on Econophysics, Budapest, July 21-27, 1997.

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., Comment: 21 pages

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the {Cauchy} problem) of the fractional diffusion equations can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to a given fractional diffusion equation., Comment: 38 pages, 8 figures (21 eps files), 1 Table

In this survey paper we consider some applications of the Wright function with special emphasis of its key role in the partial differential equations of fractional order. It was found that the Green function of the time-fractional diffusion-wave equation can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions.Finally, we discuss recent results about distribution of zeros of the Wright function, its order, type and indicator function., Comment: 34 pages. 3rd Workshop TMSF (Transform Methods and Special Functions)Sofia, Bulgaria, 1999

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour., Comment: 31 pages, 6 figures

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process., Comment: 15 pages

The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a positive measure depending on the order-distribution. We consider with some detail two cases of fractional relaxation of distributed order: the double-order and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we exhibit plots of the solutions for moderate and large times., Comment: 18 pages, 4 figures. International Symposium on Mathematical Methods in Engineering, (MME06), Ankara, Turkey, April 27-29, 2006

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative except for the single order case where the two approaches turn to be equivalent. We consider with some detail two cases of order distribution: the double-order and the uniformly distributed order. For these cases we exhibit plots of the corresponding fundamental solutions and their variance, pointing out the remarkable difference between the two approaches for small and large times., Comment: 30 pages, 4 figures. International Workshop on Fractional Differentiation and its Applications (FDA06), 19-21 July 2006, Porto, Portugal. Journal of Vibration and Control, in press (2007)

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases., Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'. Denton, Texas, August 2005

In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics. An empirical analysis performed on the 30 DJIA stocks shows that the waiting-time survival probability for high-frequency data is non-exponential. This fact imposes constraints on agent-based models of financial markets., Comment: 2 figures; preprint of a paper published on Quantitative Finance; substantially new version of an old submission (cond-mat/0310305)

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed., Comment: Preprint version of an already published paper. 8 pages

In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics. An empirical analysis performed on the 30 DJIA stocks shows that the waiting-time survival probability for high-frequency data is non-exponential. This fact sets limits for agent-based models of financial markets., Comment: 8 pages, 1 figure

The fractional diffusion equation is derived from the master equation of continuous-time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed., Comment: Paper presented at the International Workshop on Scaling and Disordered Systems, Paris, France, 13-14 April 2000

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion)., Comment: 11 pages, LaTeX

This paper reports on a collaboration between Eastern Michigan University (EMU) College of Education and Farmington High School (FHS), EMU's first consociate school, i.e., a school site that has developed an exceptionally strong working relationship with a college of education. Leaders of both institutions identified a number of creative ways they could be mutually supportive in their efforts to improve educational opportunities for young people. The consociate school relationship formally marked the beginning of a multi-year series of interrelated activities between the two institutions. So far, the team has developed a counseling program in which EMU graduate students counsel high school students and adults. In addition, a graduate course on restructuring schools for Farmington school educators was team taught by two district professionals and two EMU College of Education professors. Pre-student teaching and student teaching field experiences paired EMU students with educators from both institutions, and FHS and EMU will exchange faculty for professional development. Long range planning includes the following: improving fiber optic linkage for interactive collaborative classes; structured pre-student and student teaching experience; use of the university's professional development model to facilitate FHS's targeted goals; a proposal for integrating curriculum; a collaborative effort with Special Education, Special Olympics; and planning for the composite curriculum course to be taught on site. The following documents are included: Consociate School Partnership Agreement between Eastern Michigan University and Farmington Public Schools; a photocopy of a news story on the collaboration; a list of successes achieved, projects undertaken, and projects planned; a reprint of an article "Farmington High and Eastern Michigan Collaborate" (James R. Myers) from "Secondary Education Today: Journal of the MASSP" (v36, n2, Fall 1994); and "Farmington High School Consociate School Program Assessment Plan" (Draft). (ND)

We apply the Continuous Time Random Walk (CTRW) framework, introduced in finance by Scalas et al., to the analysis of the probability distribution of time intervals between two consecutive trades in the case of BTP futures prices traded at LIFFE in 1997. Results corroborate the validity of the CTRW approach for the description of the temporal evolution of financial time series., Comment: Submitted to Quantitative Finance, Proceedings of Application of Physics in Financial Analysis II, Liege, 13-15 July 2000

We study, both analytically and by numerical modeling the equilibrium probability density function for an non-linear L\'{e}vy oscillator with the L\'{e}vy index \alpha, 1 \leq \alpha \leq 2, and the potential energy x^4. In particular, we show that the equilibrium PDF is bimodal and has power law asymptotics with the exponent -(\alpha +3)., Comment: 14 pages, LaTeX, 8 figures gif

We complement the theory of tick-by-tick dynamics of financial markets based on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et al., and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London., Comment: Revised version, 17 pages, 4 figures. Physica A, Vol. 287, No 3-4, 468--481 (2000). Proceedings of the International Workshop on "Economic Dynamics from the Physics Point of View", Bad-Honnef (Germany), 27-30 March 2000

In this paper we present a rather general phenomenological theory of tick-by-tick dynamics in financial markets. Many well-known aspects, such as the L\'evy scaling form, follow as particular cases of the theory. The theory fully takes into account the non-Markovian and non-local character of financial time series. Predictions on the long-time behaviour of the waiting-time probability density are presented. Finally, a general scaling form is given, based on the solution of the fractional diffusion equation., Comment: 11 pages, no figures, LaTeX2e, submitted to Physica A

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index $\alpha$ ($0< \alpha \le 2$), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as Levy-Feller diffusion processes., Comment: 13 pages, 3 figures, to be published in Physica A