Marcin Magdziarz

Wroclaw University of Science and Technology | WUT · Faculty of Pure and Applied Mathematics

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or nonrandom point. Our approach is based on stochastic differential equations called jump-diffusion models. It allows to ana...

In this paper we analyze the asymptotic behavior of Lévy walks with rests. Applying recent results in the field of functional convergence of continuous-time random walks we find the corresponding limiting processes. Depending on the parameters of the model, we show that in the limit we can obtain standard Lévy walk or the process describing competi...

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem...

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black–Scholes (B–S) model. Two computational methods for valuin...

The standard model of visual search dynamics is Brownian motion. However, recent research in cognitive science reveals that standard diffusion processes seem not to be the appropriate models of human looking behavior. In particular, experimental results confirm that the superdiffusive Lévy-type dynamics appears in this context. In this paper, we an...

In this paper we focus on the tempered subdiffusive Black-Scholes (tsB-S) model. The main part of our work consists of the finite difference (FD) method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has $2...

In this paper, we investigate the relation between Bachelier and Black-Scholes (B-S) models driven by the infinitely divisible inverse subordinators. Such models, in contrast to their classical equivalents, can be used in markets where periods of stagnation are observed. We introduce the subordinated Cox-Ross-Rubinstein (CRR) model and prove that i...

Background Due to the tendency to reduce antibiotic use in humans and animals, more attention is paid to feed additives as their replacement. Crucial role of feed additives is to improve the health status, production efficiency and performance. In this original research, we estimate the potential influence of garlic (Allium sativum) extract and pro...

In this paper we study properties of the diffusion limits of three different models of Lévy walks (LW). Exact asymptotic behavior of their trajectories is found using LePage series representation. We also prove an existing conjecture about total variation of LW sample paths. Based on this conjecture we verify martingale properties of the limit proc...

In this paper we focus on the subdiffusive Black–Scholes (B–S) model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We find the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank...

In this paper we propose an extension of the Merton model. We apply the subdiffusive mechanism to analyze equity warrant in a fractional Brownian motion environment, when the short rate follows the subdiffusive fractional Black-Scholes model. We obtain the pricing formula for zero-coupon bond in the introduced model and derive the partial different...

Continuous-time random walks (CTRWs) are an elementary model for particle motion subject to randomized waiting times. In this paper, we consider the case where the distribution of waiting times depends on the location of the particle. In particular, we analyze the case where the medium exhibits a bounded trapping region in which the particle is sub...

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A c...

We investigate the first-passage dynamics of symmetric and asymmetric L\'evy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A...

This paper is focused on American option pricing in the subdiffusive Black Scholes model. Two methods for valuing American options in the considered model are proposed. The weighted scheme of the finite difference (FD) method is derived and the main properties of the method are presented. The Longstaff-Schwartz method is applied for the discussed m...

Recently, scaled Brownian motion has attracted considerable attention in the context of single particle tracking experiments displaying anomalous fractional dynamics. Its probability density function coincides with the one for fractional Brownian motion. On the other hand, scaled Brownian motion displays weak ergodicity breaking. In this paper we s...

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nic...

We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity...

Recent advances in experimental techniques for complex systems and the corresponding theoretical findings show that in many cases random parametrization of the diffusion coefficients gives adequate descriptions of the observed fractional dynamics. In this paper we introduce two statistical methods which can be effectively applied to analyze and est...

Recent results in single-particle experiments show that many complex systems display ergodicity breaking. In this paper we demonstrate how to transform a non-ergodic anomalous diffusion process in order to recover the ergodicity property. In the introduced method we use the so-called Lamperti transformation. Our approach enables us to perform stati...

In this paper we analyze asymptotic behaviour of a stochastic process called Lévy-Lorentz gas. This process is aspecial kind of continuous-time random walk in which walker moves in the fixed environment composed of scattering points. Upon each collision the walker performs a flight to the nearest scattering point. This type of dynamics is observed...

We introduce a coupled continuous-time random walk with coupling which is characteristic for Lévy walks. Additionally we assume that the walker moves in a quenched random environment, i.e. the site disorder at each lattice point is fixed in time. We analyze the scaling limit of such a random walk. We show that for large times the behaviour of the a...

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 o...

We analyze ergodic properties of two different stationary processes resulting from a combination of Lévy flights and subdiffusion. The first one comes from the Lamperti transformation of subordinated γ-stable process. The second one is a sequence of increments of subordinated Lévy process. We prove that both processes are mixing and ergodic. We als...

In this paper we generalize the classical multidimensional Black-Scholes model to the subdiffusive case. In the studied model the prices of the underlying assets follow subdiffusive multidimensional geometric Brownian motion. We derive the corresponding fractional Fokker–Plank equation, which describes the probability density function of the asset...

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 ta between the initialization and the beginning of observations. In this paper we study aging properties of ballistic Lévy walks and two closely related...

In this paper we derive explicit formulas for the densities of Lévy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represent...

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...

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...

We provide explicit formulas for asymptotic densities of d-dimensional ballistic isotropic L\'evy walks, when $d>1$. The densities of multidimensional undershooting and overshooting L\'evy walks are presented as well. Interestingly, when the number of dimensions is odd the densities of all these L\'evy walks are given by elementary functions. When...

In an article [J. Math. Phys. 53, 072701 (2012)] X. Sun and J. Duan presented Fokker-Planck equations for nonlinear stochastic differential equations with non-Gaussian L\'evy processes. In this comment we show a serious drawback in the derivation of their main result. In the proof of Theorem 1 in the aforementioned paper, a false assumption that ea...

The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of beta-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Levy W...

We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic L\'evy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D L\'evy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative which allows to...

In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given by fractional material derivative. Then we introduce a Levy walk process of distributed-order type to establis...

In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equati...

In this paper we derive explicit formulas for the densities of Levy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represent...

An efficient framework for the optimal control of the probability density function of a subdiffusion process is presented. This framework is based on a fractional Fokker–Planck equation that governs the time evolution of the PDF of the subdiffusion process and on tracking objectives of terminal configuration of the desired PDF. The corresponding op...

In this paper we analyze multidimensional Lévy walks with power-law dependence between waiting times and jumps. We obtain the detailed structure of the scaling limits of such multidimensional processes for all positive values of the power-law exponent. It appears that the scaling limit strongly depends on the value of the power-law exponent and has...

In this work we study a notion of stochastic processes which are rotationally invariant. This concept allows for simpler modelling of multi-dimensional data which exhibits rotational symmetry and is particularly useful for the treatment of systems with a complex memory structure. For this reason we introduce rotationally invariant extensions of pop...

This paper concerns the problem of large deviation for the subordinated process ZH(t)=WH(T(t))ZH(t)=WH(T(t)). The process WH={WH(t),t∈R}WH={WH(t),t∈R} is the fractional Brownian motion with Hurst index H∈(0,1)H∈(0,1) taking values in RR. T={T(t),t≥0}T={T(t),t≥0} is the inverse αα-stable subordinator. In this paper we extend the results obtained in...

The purpose of this paper is to correct errors presented recently in the paper [Lv et al. J Stat Phys 149:619-628 (2012)], where the authors analyzed Fractional Fokker-Planck equation (FFPE) with space-time dependent drift and diffusion coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representa...

We study the asymptotic behaviour of the time-changed stochastic process $\vphantom{X}^f\!X(t)=B(\vphantom{S}^f\!S (t))$, where $B$ is a standard one-dimensional Brownian motion and $\vphantom{S}^f\!S$ is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing L\'evy process with Laplace expon...

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 fu...

Progress in measurement tools and numerical simulations has led to an increase in attention towards complex systems. Recently developed methods based on the so-called p-variation appeared to be very effective in researching the stochastic origin of such phenomena, in particular in fitting the widely used fractional Brownian motion model. In this pa...

The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of application of theese processes to model certain physical phenomena. In this article we propose alternative definition of continuous time random walks with conti...

In this paper we obtain the scaling limit of a multidimensional Lévy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Lévy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method...

In this paper, we obtain the scaling limits of one-dimensional overshooting Lévy walks. We also find the limiting processes for extensions of Lévy walks, in which the waiting times and jumps are related by powerlaw, exponential and logarithmic dependence. We find that limiting processes of overshooting Lévy walk are characterized by infinite mean-s...

We study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated waiting times. In this model the current waiting time Ti is equal to the previous waiting time Ti−1 plus a small increment. Based on the associated coupled Langevin equations the force field i...

In this paper we consider a generalization of one of the earliest models of an asset price, namely the Black–Scholes model, which captures the subdiffusive nature of an asset price dynamics. We introduce the geometric Brownian motion time-changed by infinitely divisible inverse subordinators, to reflect underlying anomalous diffusion mechanism. In...

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...

In this paper we derive Langevin picture of Lévy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled Lévy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such...

Recent advances in single-molecule experiments show that various complex systems display nonergodic behavior. In this paper, we show how to test ergodicity and ergodicity breaking in experimental data. Exploiting the so-called dynamical functional, we introduce a simple test which allows us to verify ergodic properties of a real-life process. The t...

The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property is manifested by the random (infinitely divisible) periods of time, during which the asset price does...

In this chapter we propose a systematic methodology on how to distinguish between three mechanisms leading to single molecule subdiffusion, namely fractional Brownian motion, fractional Lévy stable motion and Fractional Fokker–Planck equation. We illustrate step by step that the methods of sample mean-squared displacement and p-variation can be suc...

In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by the α-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a m...

One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant c...

In this paper we introduce a Langevin-type model of subdiffusion with tempered α-stable waiting times. We consider the case of space-dependent external force fields. The model displays subdiffusive behavior for small times and it converges to standard Gaussian diffusion for large time scales. We derive general properties of tempered anomalous diffu...

In this paper, we propose a method to distinguish between mechanisms leading to single molecule subdiffusion in confinement. We show that the method of p-variation, introduced in the recent paper [M. Magdziarz, Phys. Rev. Lett. 103, 180602 (2009)], can be successfully applied also for confined systems. We propose a test which allows distinguishing...

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 subdiffus...

In this paper, we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation. We investigate properties of a coupled continuous-time random walk that follows from modeling the occurrence of jumps with compound counting processes. As a result, two different diffusion processes corresponding...

In Theorem 6 of his fundamental paper (Theory Probab. Appl., 15 (1970), pp. 1-22) Maruyama gives three necessary and sufficient conditions for a stationary infinitely divisible process to be mixing. We show that the last condition in Maruyama's theorem follows from the previous one. With this result we obtain a significantly simplified version of M...

In statistical physics, subdiffusion processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean square displacement. For the mathematical description of subdiffusion, one uses fractional Fokker–Planck equations. In this paper we construct a stochastic process, whose probability density...

In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divi...

Fractional Brownian motion with Hurst index less then 1/2 and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems. We propose a simple test, based on the analysis of the so-called p variati...

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find...

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 pro...

In this paper we study a Langevin approach to modeling of subdiffusion in the presence of time-dependent external forces. We construct a subordinated Langevin process, whose probability density function solves the subdiffusive fractional Fokker-Planck equation. We generalize the results known for the Lévy-stable waiting times to the case of infinit...

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....

A century after the celebrated Langevin paper [C.R. Seances Acad. Sci. 146, 530 (1908)] we study a Langevin-type approach to subdiffusion in the presence of time-dependent force fields. Using a subordination technique, we construct rigorously a stochastic Langevin process, whose probability density function is equal to the solution of the fractiona...

We introduce a subordination-based approach to modeling of anomalous diffusion processes in time-dependent force fields. Using the concept of inverse subordinators and the theory of Levy processes, we construct rigorously a stochastic process, which corresponds to the fractional Fokker-Planck equation with time-dependent force. Our model provides g...

Competition between subdiffusion and Lévy flights is conveniently de-scribed by the fractional Fokker–Planck equation with temporal and spatial fractional derivatives. The equivalent approach is based on the subordi-nated Langevin equation with stable noise. In this paper we examine the properties of such Langevin equation with the heavy-tailed noi...

In this paper we attack the challenging problem of modeling subdiffusion with an arbitrary space-time-dependent driving. Our method is based on a combination of the Langevin-type dynamics with subordination techniques. For the case of a purely time-dependent force, we recover the death of linear response and field-induced dispersion -- two signific...

We demonstrate that continuous-time FARIMA processes with αα-stable noise provide a new stochastic tool for studying the solar flare phenomenon in the framework of fractional Langevin equation. Simple computer tests to check the origins of αα-stability and self-similarity are implemented for empirical time series describing the energy of solar flar...

We consider five fractional generalizations of the Markovian αα-stable Ornstein–Uhlenbeck process and explore the dependence structure of these stochastic models. Since the variance of αα-stable distributed random variables is infinite, we describe the dependence structure of the introduced processes in the language of the function called codiffere...

Subdiffusion in the presence of an external force field can be described in phase space by the fractional Klein-Kramers equation. In this paper, we explore the stochastic structure of this equation. Using a subordination method, we define a random process whose probability density function is a solution of the fractional Klein-Kramers equation. The...

We investigate both analytically and numerically the first passage time (FPT) problem in one dimension for anomalous diffusion processes in which Lévy flights and subdiffusion coexist. We analyze the FPT for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index mu, 0<mu<2, and skewness parameter beta=0, (ii...

We show that, similar to the Gaussian case, the fractional Ornstein–Uhlenbeck α-stable process obtained via the Lamperti transformation of the linear fractional stable motion is a different stationary process than the one defined as the solution of the Langevin equation driven by a linear fractional stable noise. We investigate the asymptotic depen...

In this paper we answer positively a question raised by Metzler and Klafter [Phys. Rep. 339, 1 (2000)]: can one see a competition between subdiffusion and Lévy flights in the framework of the fractional Fokker-Planck dynamics? Our method of Monte Carlo simulations demonstrates the competition on the level of realizations as well as on the level of...

We introduce an approximation of the risk processes by anomalous dif-fusion. In the paper we consider the case, where the waiting times between successive occurrences of the claims belong to the domain of attraction of α-stable distribution. The relationship between the obtained approxima-tion and the celebrated fractional diffusion equation is emp...

A computer algorithm for the visualization of sample paths of anomalous diffusion processes is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anomalous diffusion in a nonconstant potential. Monte Carlo methods employing the introduced algorithm will surely provide tools for studying many...

We introduce a fractional Langevin equation with fi-stable noise and show that its so- lution fY•(t); t 2 Rg is the stationary fi-stable Ornstein-Uhlenbeck-type process recently studied in (14). We examine the asymptotic dependence structure of Y•(t) via the measure of its codependence r(µ1;µ2;t) being the difierence between the joint characteristi...

The paper presents the random-variable formalism of the anomalous diffusion processes. The emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps. We elucidate the role of the so-called inverse-time stochastic process, the main mathematical tool that allows us t...

The random-variable formalism of anomalous diffusion processes is pre-sented. We elucidate the role of the subordinate stochastic processes as the main mathematical tool that allows us to modify the dynamics of the classi-cal, exponential relaxation process. In particular, we discuss the anomalous diffusion schemes underlying the stretched exponent...

Dedicated to Professor Andrzej Fuliński on the occasion of his 70th birthday We show how to modify the random-walk scenario underlying the clas-sical, exponential relaxation response in order to derive the empirical Havriliak–Negami function, commonly used to fit the dielectric permit-tivity of complex-material data. The turnover from the exponenti...