Comment on "Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian L\'evy processes" [J. Math. Phys. 53, 072701 (2012)]

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 each infinitely differentiable function with compact support is equal to its Taylor series, is used. We prove that although the derivation is incorrect, the result remains valid only if we add certain additional assumptions

Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, Levy noise and space-time-dependent coefficients

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 equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality

Method of calculating densities for isotropic L\'evy Walks

We provide explicit formulas for asymptotic densities of $d$-dimensional 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 $d$ is even, we can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible

Explicit Densities of Multidimensional L\'evy Walks

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 efficiently evaluate them numerically. The theoretical results agree with Monte-Carlo simulations. The obtained functions solve certain differential equations with the fractional material derivative

Limit theorems for continuous time random walks with continuous paths

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 continuous trajectories. We also give the functional limit theorem for sequence of such random walks. This result requires the use of strong Skorohod M1 topology instead of Skorohod J1 topology, which is usually used in limit theorems for ordinary CTRW processes

Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

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 establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with material derivative of distributed-order type

Asymptotic properties of Brownian motion delayed by inverse subordinators

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 exponent $f$. This type of processes plays an important role in statistical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffusion process. We also investigate various martingale properties, derive a generalized Feynman-Kac formula, the laws of large numbers and of the iterated logarithm for $\vphantom{X}^f\!X$

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.Comment: 40 pages, 7 figure

Codifference can detect ergodicity breaking and non-Gaussianity

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 models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a non-linear analogue of the mean squared displacement.Comment: 39 pages, 5 figures, IOP LaTe

A weighted finite difference method for subdiffusive Black Scholes Model

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-Nicolson scheme. The proposed method has $2-\alpha$ order of accuracy with respect to time where $\alpha\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results