@article{PetreskaPejovSandevetal.2022, author = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf}, title = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {2}, publisher = {MDPI AG, Fractal Fract Editorial Office}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6020088}, pages = {23}, year = {2022}, abstract = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.}, language = {en} } @article{StojkoskiSandevBasnarkovetal.2020, author = {Stojkoski, Viktor and Sandev, Trifce and Basnarkov, Lasko and Kocarev, Ljupco and Metzler, Ralf}, title = {Generalised geometric Brownian motion}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {12}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22121432}, pages = {34}, year = {2020}, abstract = {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 investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions 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) in order 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.}, language = {en} } @article{SandevIominKocarev2020, author = {Sandev, Trifce and Iomin, Alexander and Kocarev, Ljupco}, title = {Hitting times in turbulent diffusion due to multiplicative noise}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Institute of Physics}, address = {Woodbury, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.042109}, pages = {10}, year = {2020}, abstract = {We study a distribution of times of the first arrivals to absorbing targets in turbulent diffusion, which is due to a multiplicative noise. Two examples of dynamical systems with a multiplicative noise are studied. The first one is a random process according to inhomogeneous diffusion, which is also known as a geometric Brownian motion in the Black-Scholes model. The second model is due to a random processes on a two-dimensional comb, where inhomogeneous advection is possible only along the backbone, while Brownian diffusion takes place inside the branches. It is shown that in both cases turbulent diffusion takes place as the one-dimensional random process with the log-normal distribution in the presence of absorbing targets, which are characterized by the Levy-Smirnov distribution for the first hitting times.}, language = {en} } @article{BasnarkovTomovskiSandevetal.2022, author = {Basnarkov, Lasko and Tomovski, Igor and Sandev, Trifce and Kocarev, Ljupčo}, title = {Non-Markovian SIR epidemic spreading model of COVID-19}, series = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, volume = {160}, journal = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, publisher = {Elsevier}, address = {Oxford [u.a.]}, issn = {0960-0779}, doi = {10.1016/j.chaos.2022.112286}, pages = {8}, year = {2022}, abstract = {We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete-and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.}, language = {en} } @article{SandevDomazetoskiKocarevetal.2022, author = {Sandev, Trifce and Domazetoski, Viktor and Kocarev, Ljupco and Metzler, Ralf and Chechkin, Aleksei}, title = {Heterogeneous diffusion with stochastic resetting}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {7}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac491c}, pages = {26}, year = {2022}, abstract = {We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation.}, language = {en} } @article{PengSandevKocarev2021, author = {Peng, Junhao and Sandev, Trifce and Kocarev, Ljupco}, title = {First encounters on Bethe lattices and Cayley trees}, series = {Communications in nonlinear science \& numerical simulation}, volume = {95}, journal = {Communications in nonlinear science \& numerical simulation}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1007-5704}, doi = {10.1016/j.cnsns.2020.105594}, pages = {15}, year = {2021}, abstract = {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 probabilities (SPs) of the target A on the both kinds of structures are considered 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. (C) 2020 Elsevier B.V. All rights reserved.}, language = {en} } @article{StojkoskiJolakoskiPaletal.2022, author = {Stojkoski, Viktor and Jolakoski, Petar and Pal, Arnab and Sandev, Trifce and Kocarev, Ljupco and Metzler, Ralf}, title = {Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity}, series = {Philosophical transactions of the Royal Society A: Mathematical, physical and engineering sciences}, volume = {380}, journal = {Philosophical transactions of the Royal Society A: Mathematical, physical and engineering sciences}, number = {2224}, publisher = {Royal Society}, address = {London}, issn = {1364-503X}, doi = {10.1098/rsta.2021.0157}, pages = {17}, year = {2022}, abstract = {We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and income mobility, the feasibility of an individual to change their position in the income rankings. For this purpose, we use the properties of an established model for income growth that includes 'resetting' as a stabilizing force to ensure stationary dynamics. We find that the dynamics of inequality is regime-dependent: it may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodicity. Mobility measures, conversely, are always stable over time, but suggest that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the income share of the top earners in the USA, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterizes inequality and immobility. Our results can serve as a simple rationale for the observed real-world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe.This article is part of the theme issue 'Kinetic exchange models of societies and economies'.}, language = {en} } @article{StojkoskiSandevKocarevetal.2022, author = {Stojkoski, Viktor and Sandev, Trifce and Kocarev, Ljupco and Pal, Arnab}, title = {Autocorrelation functions and ergodicity in diffusion with stochastic resetting}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac4ce9}, pages = {22}, year = {2022}, abstract = {Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift-diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.}, language = {en} }