Marcin Magdziarz

Path Properties of Subdiffusion—A Martingale Approach

In statistical physics, subdiffusion processes constitute one of the most relevant subclasses of the family of anomalous diffusion models. These processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean-squared displacement. In this article we study sample path properties of subdiffusion. We propose a martingale approach to the stochastic analysis of subdiffusion models. We verify the martingale property, Hölder continuity of the trajectories, and derive the law of large numbers. The precise asymptotic behavior of subdiffusion is obtained in the law of the iterated logarithm. The presented results may be applied to identify the type of subdiffusive dynamics in experimental data.

FARIMA MODELING OF SOLAR FLARE ACTIVITY FROM EMPIRICAL TIME SERIES OF SOFT X-RAY SOLAR EMISSION

A time series of soft X-ray emission observed by the Geostationary Operational Environment Satellites from 1974 to 2007 is analyzed. We show that in the solar-maximum periods the energy distribution of soft X-ray solar flares for C, M, and X classes is well described by a fractional autoregressive integrated moving average model with Pareto noise. The model incorporates two effects detected in our empirical studies. One effect is a long-term dependence (long-term memory), and another corresponds to heavy-tailed distributions. The parameters of the model: self-similarity exponent H, tail index α, and memory parameter d are statistically stable enough during the periods 1977-1981, 1988-1992, 1999-2003. However, when the solar activity tends to minimum, the parameters vary. We discuss the possible causes of this evolution and suggest a statistically justified model for predicting the solar flare activity.

Multidimensional Levy walk and its scaling limits

In this paper we obtain the scaling limit of a multidimensional Levy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Levy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Levy walks with a predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data—we are able to recover the spectral measure from the data and obtain the full characterization of a Levy walk. We also give examples of some useful spectral measures, which cover a large class of possible scenarios in the modeling of real-life phenomena.

Anomalous diffusion and semimartingales

We argue that the essential part of the currently explored models of anomalous (non-Brownian) diffusion are actually Brownian motion subordinated by the appropriate random time. Thus, in many cases, anomalous diffusion can be embedded in Brownian diffusion. Such an embedding takes place if and only if the anomalous diffusion is a semimartingale process. We also discuss the structure of anomalous diffusion models. Categorization of semimartingales can be applied to differentiate among various anomalous processes. In particular, identification of the type of subdiffusive dynamics from experimental data is feasible.

Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs

This study investigates a new formula for option pricing with transaction costs in a discrete time setting. The value of the financial assets is based on time-changed mixed fractional Brownian motion (MFBM) model. The pricing method is obtained for European call option using the time-changed MFBM model in a discrete time setting. Particularly, the minimal value Cmin(t,St) of an option respect to transaction costs is obtained. Furthermore, the new model for pricing currency option is presented by utilizing the time-changed MFBM model. In addition, the impact of time step Δt, Hurst parameter H and transaction costs α are also investigated, which substantiate that these parameters play a significant role in our pricing formula. Finally, the empirical studies and the simulation findings corroborate the theoretical bases and indicate the time-changed MFBM is a satisfactory model.

Asymptotic behaviour of random walks with correlated temporal structure

We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.

Correlated continuous-time random walks?scaling limits and Langevin picture

In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.

Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analytical analysis demonstrating how various types of parameter distributions for the memory kernel result in the exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in relaxation from Gaussian to non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with Monte Carlo simulations.

Aging ballistic Lévy walks

Aging can be observed for numerous physical systems. In such systems statistical properties [like probability distribution, mean square displacement (MSD), first-passage time] depend on a time span t_{a} between the initialization and the beginning of observations. In this paper we study aging properties of ballistic Lévy walks and two closely related jump models: wait-first and jump-first. We calculate explicitly their probability distributions and MSDs. It turns out that despite similarities these models react very differently to the delay t_{a}. Aging weakly affects the shape of probability density function and MSD of standard Lévy walks. For the jump models the shape of the probability density function is changed drastically. Moreover for the wait-first jump model we observe a different behavior of MSD when t_{a}≪t and t_{a}≫t.

Explicit densities of multidimensional ballistic Lévy walks.

Lévy walks have proved to be useful models of stochastic dynamics with a number of applications in the modeling of real-life phenomena. In this paper we derive explicit formulas for densities of the two- (2D) and three-dimensional (3D) ballistic Lévy walks, which are most important in applications. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D Lévy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative, which allows us to efficiently evaluate them numerically. The theoretical results agree perfectly with Monte Carlo simulations.