arXiv subjects

Trifce Sandev

Publications and source records attributed to Trifce Sandev.

36 records · Page 2Linked to original sources

arXiv2020

Continuous time random walks under Markovian resetting

We investigate the effects of markovian resseting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power law probability density functions. We prove the existence of a non-equilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target and the characteristic transport exponents.

cond-mat.stat-mech↗

arXiv2020

Generalised geometric Brownian motion: Theory and applications to option pricing

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics due to irregularities found when comparing its properties with empirical distributions. As a solution, we develop a generalisation of GBM where the introduction of a memory kernel critically determines the behavior of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and obtain the corresponding probability density functions by using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

q-fin.PR↗

arXiv2020

First encounters on Bethe Lattices and Cayley Trees

In this work we consider the first encounter problems between a fixed and/or mobile target A and a moving trap B on Bethe Lattices and Cayley trees. The survival probability (SP) of the target A on the both kinds of structures are analyzed analytically and compared. On Bethe Lattices, the results show that the fixed target will still prolong its survival time, whereas, on Cayley trees, there are some initial positions where the target should move to prolong its survival time. The mean first encounter time (MFET) for mobile target A is evaluated numerically and compared with the mean first passage time (MFPT) for the fixed target A. Different initial settings are addressed and clear boundaries are obtained. These findings are helpful for optimizing the strategy to prolong the survival time of the target or to speed up the search process on Cayley trees, in relation to the target's movement and the initial position configuration of the two walkers. We also present a new method, which uses a small amount of memory, for simulating random walks on Cayley trees.

cond-mat.stat-mech↗

arXiv2020

Stochastic resetting on comb-like structures

We study a diffusion process on a three-dimensional comb under stochastic resetting. We consider three different types of resetting: global resetting from any point in the comb to the initial position, resetting from a finger to the corresponding backbone and resetting from secondary fingers to the main fingers. The transient dynamics along the backbone in all three cases is different due to the different resetting mechanisms, finding a wide range of dynamics for the mean squared displacement. For the particular geometry studied herein, we compute the stationary solution and the mean square displacement and find that the global resetting breaks the transport in the three directions. Regarding the resetting to the backbone, the transport is broken in two directions but it is enhanced in the main axis. Finally, the resetting to the fingers enhances the transport in the backbone and the main fingers but reaches a steady value for the mean squared displacement in the secondary fingers.

cond-mat.stat-mech↗

arXiv2019

Lévy walk revisited: Hermite polynomial expansion approach

Integral transform method (Fourier or Laplace transform, etc) is more often effective to do the theoretical analysis for the stochastic processes. However, for the time-space coupled cases, e.g., Lévy walk or nonlinear cases, integral transform method may fail to be so strong or even do not work again. Here we provide Hermite polynomial expansion approach, being complementary to integral transform method. Some statistical observables of general Lévy walks are calculated by the Hermite polynomial expansion approach, and the comparisons are made when both the integral transform method and the newly introduced approach work well.

cond-mat.stat-mech↗

arXiv2019

Generalized diffusion-wave equation with memory kernel

We study generalized diffusion-wave equation in which the second order time derivative is replaced by integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.

cond-mat.stat-mech↗

arXiv2019

Generalised Diffusion and Wave Equations: Recent Advances

We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional, distributed order, and tempered time-fractional diffusion and wave equations. Such equations exhibit multi-scaling time behaviour, which makes them suitable for the description of different diffusive regimes and characteristic crossover dynamics in complex systems.

cond-mat.stat-mech↗

arXiv2018

Finite-velocity diffusion on a comb

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox $H$-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox $H$-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.

cond-mat.stat-mech↗

arXiv2018

Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.

cond-mat.stat-mech↗

arXiv2017

Heterogeneous diffusion in comb and fractal grid structures

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-α)}$, where $α$ is the power-law exponent of the position dependent diffusion coefficient $D(x)\sim |x|^α$. Depending on the value of $α$ we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+ν)/(2-α)}$, where $0<ν<1$ is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox $H$-functions.

cond-mat.stat-mech↗

arXiv2016

Anomalous diffusion on a fractal mesh

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2(t)\rangle\sim t^β$, where the transport exponent $β<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $β>1$ has been observed as well when the environment is controlled by means of a memory kernel.

cond-mat.stat-mech↗

arXiv2016

Generalized distributed order diffusion equations with composite time fractional derivative

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox $H$-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper.

math-ph↗

arXiv2016

Lévy Transport in Slab Geometry of Inhomogeneous Media

We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.

quant-ph↗

arXiv2016

Lévy processes on a generalized fractal comb

Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structure can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Lévy processes (Lévy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Lévy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox $H$-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractional structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. It is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.

cond-mat.stat-mech↗

arXiv2016

Comb model with slow and ultraslow diffusion

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyse the probability distribution functions and we derive the mean squared displacement in $x$ and $y$ directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both $x$ and $y$ directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.

cond-mat.stat-mech↗

arXiv2015

Fractional diffusion on a fractal grid comb

A grid comb model is a generalization of the well known comb model, and it consists of $N$ backbones. For $N=1$ the system reduces to the comb model where subdiffusion takes place with the transport exponent $1/2$. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.

cond-mat.stat-mech↗

arXiv2014

Fractional Helmholtz and fractional wave equations with Riesz-Feller and generalized Riemann-Liouville fractional derivatives

The objective of this paper is to derive analytical solutions of fractional order Laplace, Poisson and Helmholtz equations in two variables derived from the corresponding standard equations in two dimensions by replacing the integer order partial derivatives with fractional Riesz-Feller derivative and generalized Riemann-Liouville fractional derivative recently defined by Hilfer. The Fourier-Laplace transform method is employed to obtain the solutions in terms of Mittag-Leffler functions, Fox $H$-function and an integral operator containing a Mittag-Leffler function in the kernel. Results for fractional wave equation are presented as well. Some interesting special cases of these equations are considered. Asymptotic behavior and series representation of solutions are analyzed in detail. Many previously obtained results can be derived as special cases of those presented in this paper.

math-ph↗

arXiv2011

Delayed feedback control of fractional-order chaotic systems

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parametrizad by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of the steady states in comparison to the classical time-delayed feedback scheme with a constant delay.

physics.gen-ph↗