Search Results (34)

This paper studies Dirichlet problems for one-dimensional Lévy-type nonlocal elliptic equations on the half-line. The equation $L^{\mu }\nu (x)=f(x),x>0,\nu (x)=0,x\le 0$ is transformed into a weighted nonlocal equation associated with a multiplicative jump process. Under basic structural assumptions on the Lévy measure, the transformed generator is realized through a martingale problem, and the associated exponential killing representation gives a probabilistic mild solution with an immediate $L^{\infty }$-estimate. For the one-dimensional fractional Laplacian, the transformed process is exactly multiplicative. This yields a new approach in which solution estimates are derived from the stochastic equation of the transformed process; smooth-data resolvent solutions are estimated in weighted $L_p$-spaces and extended to general data by approximation. For more general Lévy measures, a smooth weighted energy estimate is proved. The key analytic input is a weighted adjoint integral inequality for the transformed generator, verified for subordinate Brownian motions associated with Bernstein functions and for non-unimodal logarithmically perturbed stable-type operators. Full article

Each charging/discharging cycle leads to a gradual decrease in the battery’s capacity. The degradation of capacity in lithium-ion batteries represents a non-monotonous process with random jumps. Earlier studies claimed that the instantaneous degradation value of a lithium-ion battery is influenced by the historical dataset with long-range dependence. The existing methods ignore large jumps and long-range dependences in degradation processes. In order to capture long-range-dependent behavior with random jumps, we refer to the fractional Poisson process. We also outline the relationship between the long-range correlation and the Hurst index. The connection between random jumps in capacitance and long-range dependence of the fractional Poisson process is proven. In order to construct the fractional Poisson predictive model, we included fractional Brownian motion as the diffusion term and the fractional Poisson process as the jump term. The proposed approach is implemented on NASA’s dataset for Li-ion battery degradation. We believe that the error analysis for the fractional Poisson process is advantageous compared with that of the fractional Brownian motion, the fractional Levy stable motion, the Wiener model, and the long short-term memory model. Full article

In some problems, partial differential equations are reduced to ordinary differential equations. In special cases, when incorporating randomness, equations can be reduced to systems of stochastic differential Equations (SDEs). Stochastic averaging for a class of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise is considered. Stability criteria for systems of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise do not currently exist. Usually, studies on determining the sensitivity of solutions to the accuracy of setting the initial conditions are being conducted to explain the phenomenon of deterministic chaos. These studies show both convergence in mean square and convergence in probability to averaged systems of stochastic differential equations driven by fractional Brownian motion and Lévy process. The solutions to systems can be approximated by solutions to averaged stochastic differential equations by using the stochastic averaging. Full article

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory. Full article

We consider two different time fractional telegrapher’s equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates due to the resetting mechanism. We also obtain the fractional telegraph process as a subordinated telegraph process by introducing operational time such that the physical time is considered as a Lévy stable process whose characteristic function is the Lévy stable distribution. We also analyzed the survival probability for the first-passage time problem and found the optimal resetting rate for which the corresponding mean first-passage time is minimal. Full article

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers. Full article

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-$\alpha$-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. Full article

In this study, we present a novel approach to estimating the Hurst exponent of time series data using a variety of machine learning algorithms. The Hurst exponent is a crucial parameter in characterizing long-range dependence in time series, and traditional methods such as Rescaled Range (R/S) analysis and Detrended Fluctuation Analysis (DFA) have been widely used for its estimation. However, these methods have certain limitations, which we sought to address by modifying the R/S approach to distinguish between fractional Lévy and fractional Brownian motion, and by demonstrating the inadequacy of DFA and similar methods for data that resembles fractional Lévy motion. This inspired us to utilize machine learning techniques to improve the estimation process. In an unprecedented step, we train various machine learning models, including LightGBM, MLP, and AdaBoost, on synthetic data generated from random walks, namely fractional Brownian motion and fractional Lévy motion, where the ground truth Hurst exponent is known. This means that we can initialize and create these stochastic processes with a scaling Hurst/scaling exponent, which is then used as the ground truth for training. Furthermore, we perform the continuous estimation of the scaling exponent directly from the time series, without resorting to the calculation of the power spectrum or other sophisticated preprocessing steps, as done in past approaches. Our experiments reveal that the machine learning-based estimators outperform traditional R/S analysis and DFA methods in estimating the Hurst exponent, particularly for data akin to fractional Lévy motion. Validating our approach on real-world financial data, we observe a divergence between the estimated Hurst/scaling exponents and results reported in the literature. Nevertheless, the confirmation provided by known ground truths reinforces the superiority of our approach in terms of accuracy. This work highlights the potential of machine learning algorithms for accurately estimating the Hurst exponent, paving new paths for time series analysis. By marrying traditional finance methods with the capabilities of machine learning, our study provides a novel contribution towards the future of time series data analysis. Full article

The PHENIX experiment measured two-particle Bose–Einstein quantum-statistical correlations of charged kaons in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The correlation functions are parametrized assuming that the source emitting the particles has a Lévy shape, characterized by the Lévy exponent $\alpha$ and the Lévy scale R. By introducing the intercept parameter $\lambda$, we account for the core–halo fraction. The parameters are investigated as a function of transverse mass. The comparison of the parameters measured for kaon–kaon with those measured from pion–pion correlation may clarify the connection of Lévy parameters to physical processes. Full article

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here, we derive the corresponding limiting equations in the case of position-depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo–Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalizations are analyzed. Full article

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data. Full article

The paper builds a Variance-Gamma (VG) model with five parameters: location ($\mu$), symmetry ($\delta$), volatility ($\sigma$), shape ($\alpha$), and scale ($\theta$); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a $\mathsf{\Gamma }(\alpha ,\theta )$ Ornstein–Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order $\nu =0$, intensity $\alpha$, and steepness parameters $\frac{\delta }{{\sigma }^2}-\sqrt{\frac{{\delta }^2}{{\sigma }^4}+\frac{2}{\theta {\sigma }^2}}$ and $\frac{\delta }{{\sigma }^2}+\sqrt{\frac{{\delta }^2}{{\sigma }^4}+\frac{2}{\theta {\sigma }^2}}$; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean $(\mu +\alpha \theta \delta )$ and variance $\alpha ({\theta }^2{\delta }^2+{\sigma }^2\theta )$. The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton–Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black–Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options. Full article

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process. Full article

In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by a fractional Lévy process. The option price is given by a deterministic representation by means of a real valued function satisfying some fractional PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted Grunwald approximation. Full article

Since its inception in 2009, Bitcoin has increasingly gained main stream attention from the general population to institutional investors. Several models, from GARCH type to jump-diffusion type, have been developed to dynamically capture the price movement of this highly volatile asset. While fitting the Gaussian and the Generalized Hyperbolic and the Normal Inverse Gaussian (NIG) distributions to log-returns of Bitcoin, NIG distribution appears to provide the best fit. The time-varying Hurst parameter for Bitcoin price reveals periods of randomness and mean-reverting type of behaviour, motivating the study in this paper through fractional Ornstein–Uhlenbeck driven by a Normal Inverse Gaussian Lévy process. Features such as long-range memory are jump diffusion processes that are well captured with this model. The results present a 95% prediction for the price of Bitcoin for some specific dates. This study contributes to the literature of Bitcoin price forecasts that are useful for Bitcoin options traders. Full article