Search Results (6)

Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time series’s next value by analysing its past. The input financial data are assumed to be consistent with an $\alpha$-stable Lévy motion. The computation of the scaling exponent and the value of $\alpha$, which characterises the type of the $\alpha$-stable Lévy motion, are crucial for the iterative scheme. These two indices can be determined at each iteration from the form of the structure function, for the computation of which we use the method of generalised moments. Their values are used for the creation of the corresponding $\alpha$-stable Lévy noise, which acts as a seed for the stochastic component. Furthermore, the drift and diffusion terms are calculated at each iteration. The proposed model is general, allowing the kind of stochastic process to vary from one iterative step to another, and its applicability is not restricted to financial data. As a case study, we consider Greece’s stock market general index over a period of five years, from September 2019 to September 2024, after the completion of bailout programmes. Greece’s economy changed from a restricted to a free market over the chosen era, and its stock market trading increments are likely to be describable by an $\alpha$-stable L’evy motion. We find that $\alpha =2$ and the scaling exponent H varies over time for every iterative step we perform. The forecasting points follow the same trend, are in good agreement with the actual data, and for most of the forecasts, the percentage error is less than 2%. Full article

A common approach to simulating a Lévy process is to truncate its shot-noise representation. We focus on subordinators and introduce the remainder process, which represents the jumps that are removed by the truncation. We characterize when these processes are self-similar and show that, in the self-similar case, they can be indexed by a parameter $\alpha \in (-\infty ,1)$. When $\alpha \in (0,1)$, they correspond to $\alpha$-stable distributions, and when $\alpha =0$, they correspond to certain generalizations of the Dickman distribution. Thus, the Dickman distribution plays the role of a 0-stable distribution in this context. Full article

In this paper, we consider a class of stochastic differential equations driven by multiplicative $\alpha$-stable ($0<\alpha <2$) Lévy noises. Firstly, we show that there exists a unique strong solution under a local one-sided Lipschitz condition and a general non-explosion condition. Next, the weak Feller and stationary properties are derived. Furthermore, a concrete sufficient condition for the coefficients is presented, which is different from the conditions for SDEs driven by Brownian motion or general squared-integrable martingales. Finally, some ergodic and mixing properties are obtained by using the Foster–Lyapunov criteria. Full article

Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, $\alpha$-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value ${\mathfrak{R}}_0^{🟉}$ is calculated. For strictly positive value of ${\mathfrak{R}}_0^{🟉}$, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of ${\mathfrak{R}}_0^{🟉}$, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered $\alpha$-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter $\alpha$ have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function. Full article

The noise that is associated with nonequilibrium processes commonly features more outliers and is therefore often taken to be Lévy noise. For a Langevin particle that is subjected to Lévy noise, the kicksizes are drawn not from a Gaussian distribution, but from an $\alpha$-stable distribution. For a Gaussian-noise-subjected particle in a potential well, microscopic reversibility applies. However, it appears that the time-reversal-symmetry is broken for a Lévy-noise-subjected particle in a potential well. Major obstacles in the analysis of Langevin equations with Lévy noise are the lack of simple analytic formulae and the infinite variance of the $\alpha$-stable distribution. We propose a measure for the violation of time-reversal symmetry, and we present a procedure in which this measure is central to a controlled imposing of time-reversal asymmetry. The procedure leads to behavior that mimics much of the effects of Lévy noise. Our imposing of such nonequilibrium leads to concise analytic formulae and does not yield any divergent variances. Most importantly, the theory leads to simple corrections on the Fluctuation–Dissipation Relation. Full article

The standard textbooks contain good explanations of how and why equilibrium thermodynamics emerges in a reservoir with particles that are subjected to Gaussian noise. However, in systems that convert or transport energy, the noise is often not Gaussian. Instead, displacements exhibit an $\alpha$-stable distribution. Such noise is commonly called Lévy noise. With such noise, we see a thermodynamics that deviates from what traditional equilibrium theory stipulates. In addition, with particles that can propel themselves, so-called active particles, we find that the rules of equilibrium thermodynamics no longer apply. No general nonequilibrium thermodynamic theory is available and understanding is often ad hoc. We study a system with overdamped particles that are subjected to Lévy noise. We pick a system with a geometry that leads to concise formulae to describe the accumulation of particles in a cavity. The nonhomogeneous distribution of particles can be seen as a dissipative structure, i.e., a lower-entropy steady state that allows for throughput of energy and concurrent production of entropy. After the mechanism that maintains nonequilibrium is switched off, the relaxation back to homogeneity represents an increase in entropy and a decrease of free energy. For our setup we can analytically connect the nonequilibrium noise and active particle behavior to entropy decrease and energy buildup with simple and intuitive formulae. Full article