Rudolf Gorenflo

Fractional calculus and continuous-time finance

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 Levy 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.

On Mittag-Leffler-type functions in fractional evolution processes

Abstract We review a variety of fractional evolution processes (so defined being governed by equations of fractional order), whose solutions turn out to be related to Mittag-Leffler-type functions. The chosen equations are the simplest of the fractional calculus and include the Abel integral equations of the second kind, which are relevant in typical inverse problems, and the fractional differential equations, which govern generalized relaxation and oscillation phenomena.

Time Fractional Diffusion: A Discrete Random Walk Approach

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.

Discrete random walk models for space-time fractional diffusion

Abstract 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. By space–time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2] and skewness θ (|θ|⩽min{α,2−α}), and the first-order time derivative with a Caputo derivative of order β∈(0,1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation 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 this fractional diffusion equation.

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike 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, cannot 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 their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

Wright functions as scale-invariant solutions of the diffusion-wave equation

The time-fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0 <α ≤ 2). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional diffusion-wave equation are given in terms of the Wright function in the case 0 <α< 1 and in terms of the generalized Wright function in the case 1 <α< 2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo type modification of the

Mittag-Leffler Functions, Related Topics and Applications

As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control and several other related areas.

Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit

A proper transition to the so-called diffusion or hydrodynamic limit is discussed for continuous time random walks. It turns out that the probability density function for the limit process obeys a fractional diffusion equation. The relevance of these results for financial applications is briefly discussed.

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks. 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.

Fractional diffusion in inhomogeneous media

Starting from the continuous time random walk (CTRW) scheme with the space-dependent waiting-time probability density function (PDF) we obtain the time-fractional diffusion equation with varying in space fractional order of time derivative. As an example, we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non-trivial drift and subdiffusion whose laws are changed in the course of time.