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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

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 ct al. (Physica A 284 (2000) 376), and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London

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).

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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. REMARK: Concerning the inversion of the Riesz potential operator I α 0 let us point out that its common hyper-singular integral representation fails for α = 1. In our Section 2 we have shown that the corresponding hyper-singular representation for the inverse operator D α 0 can be obtained also in the critical (often excluded) case α = 1, by analytic continuation.

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. Furthermore, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

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