We apply the formalism of the continuous-time random walk to the study of financial data. The entire distribution of prices can be obtained once two auxiliary densities are known. These are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump. We have applied the formalism to data on the U.S. dollar-deutsche mark future exchange, finding good agreement between theory and the observed data.

Abstract.In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Inspiring properties and achievementsIn their pioneering work published in year 1965 [1], physicists Eliott W. Montroll and George H. Weiss introduced the concept of continuous-time random walk (CTRW) as a way to make the interevent-time continuous and fluctuating. It is characterized by some distribution associated with a stochastic process, giving an insight into the process activity. This distribution, called pausing-or waiting-time one (WTD), permitted the description of both Debye (exponential) and, what is most significant, non-Debye (slowly-decaying) relaxations as well as normal and anomalous transport and diffusion [2,3] -thus the model involves fundamental aspects of the stochastic world -a real, complex world. Notably, ancestors of this concept are presented by Michael Shlesinger in [4].Let us incidentally comment that term "walk" in the name "continuous-time random walk" is commonly used in the generic sense comprising two concepts: namely, both the walk (associated with finite displacement velocity of the process) and flight (associated with an instantaneous displacement of the process). Thus we have to specify in a detailed way with what kind of process we are considering.The CTRW formalism was most conveniently developed by physicists Scher and Lax in terms of recursion relations [5][6][7][8][9][10]. In this context the distinction between Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver. a e-mail: ryszard.kutner@fuw.edu.pl discrete and continuous times [11] and also between separable and non-separable WTDs were introduced [12]. A thorough analysis of the latter, called also the nonindependent CTRW, was performed a decade ago [13] although, in the context of the concentrated lattice gases, it was performed much earlier [14,15]. These analyses took into account dependences over many correlated consecutive particle displacements and waiting (or interevent) times. The Scher and Lax formulation of the CTRW formalism is particularly convenient to study as well anomalous transport and diffusion as the non-Debye relaxation and their anomalous scaling properties (e.g., the nonlinear time growth of the process variance). Examples are the span of the walk, the first-passage times, survival probabilities, the number of distinct sites visited and, of course, mean and mean-square displacement if they exist. It is very interesting...

The phenomenon of diffusion modelled as Brownian motion or in terms of the diffusion equation is deficient in that it permits an infinite speed of signal propagation. The simplest diffusion-like equation with a bounded speed of signal propagation is the so-called telegrapher's equation which implicitly contains a type of momentum. We discuss some properties, solutions and problems related to this equation. These are generally more complicated in form than corresponding ones for the diffusion equation, exhibiting wave-like properties at short times and diffusion-like properties in the long-time limit. Resumen. El fenómeno de la difusión, modelado tanto como movimiento Browniano como mediante la ecuación de difusión, es deficiente en el sentido que permite una velocidad de propagación infinita. La denominada ecuación del telegrafista es una de las ecuaciones más simples con una velocidad de propagación acotada y que contiene de forma implícita un cierto tipo momento. En este trabajo se discuten algunas propiedades, soluciones y cuestiones relacionadas con esta ecuación. Estas propiedades y soluciones son generalmente más complicadas que las correspondientes a la ecuación de difusión ordinaria, presentando, a tiempos cortos, propiedades de tipo ondulatorio y propiedades difusivas en el límite asintótico.

We study the exponential Ornstein-Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behavior in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.

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