Search Results (56)

For gas source localization in three-dimensional space, the leakage diffusion of the gas source is first modeled as a superposition of multiple Gaussian puffs. An improved Fireworks Optimization Algorithm is then proposed, which incorporates elements from the Grey Wolf Optimization algorithm and the Fireworks Algorithm. The localization process is divided into two stages: global localization and local localization. The global localization stage integrates the GWO algorithm with Lévy flight to facilitate global exploration in three-dimensional space. The local localization stage enhances the search operators and selection strategies of the Fireworks Algorithm to perform local exploitation based on the results of the global localization. To investigate the applicability of the Grey Wolf Optimization algorithm and the Fireworks Algorithm, their performance is tested in noisy environments and with different sensor arrays. The experimental results indicate that the algorithms perform well under Gaussian noise with a variance of 0.4 and with more than nine valid sensors. Furthermore, a comparison is made between the Grey Wolf Optimization algorithm, the Fireworks Algorithm, and Particle Swarm Optimization. The simulation experiments demonstrate that the proposed algorithm exhibits greater stability, enhanced computational efficiency, and reduced randomness compared to both the Grey Wolf Optimization algorithm and Particle Swarm Optimization. It requires fewer valid sensor measurements and achieves higher accuracy in scenarios with limited valid sensor information. Overall, in the context of gas source localization, it outperforms both the Grey Wolf Optimization algorithm and Particle Swarm Optimization. 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

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

This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. Then, we explore a range of financial applications, spanning scenarios characterised by the absence of jumps, the presence of jumps following a log-normal distribution, and jumps following a distribution of greater generality. Additionally, we delve into optimising more complex portfolios composed of multiple risky assets alongside a risk-free asset, shedding new light on optimal allocation strategies in these settings. Our investigation yields novel insights and potentially groundbreaking results, offering fresh perspectives on portfolio management strategies under jump-diffusion dynamics. Full article

Accurate crude oil price forecasting is essential, considering oil’s critical role in the global economy. However, the crude oil market is significantly influenced by external, transient events, posing challenges in capturing price fluctuations’ complex dynamics and uncertainties. Traditional time series forecasting models, such as ARIMA and LSTM, often rely on assumptions regarding data structure, limiting their flexibility to estimate volatility or account for external shocks effectively. Recent research highlights Generative Adversarial Networks (GANs) as a promising alternative approach for capturing intricate patterns in time series data, leveraging the adversarial learning framework. This paper introduces a Crude Oil-Driven Conditional GAN (CO-CGAN), a hybrid model for enhancing crude oil price forecasting by combining advanced AI frameworks (GANs), oil market sentiment analysis, and stochastic jump-diffusion models. By employing conditional supervised training, the inherent structure of the data distribution is preserved, thereby enabling more accurate and reliable probabilistic price forecasts. Additionally, the CO-CGAN integrates a Lévy process and sentiment features to better account for uncertainties and price shocks in the crude oil market. Experimental evaluations on two real-world oil price datasets demonstrate the superior performance of the proposed model, achieving a Mean Squared Error (MSE) of 0.000054 and outperforming benchmark models. Full article

Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts for the observed non-Gaussian distributions, long-range correlations and memory effects in cosmic ray fluxes. We derive analytical solutions using the Adomian Decomposition Method and express them in terms of Mittag-Leffler functions and Lévy stable distributions. The model parameters, including the fractional orders $\alpha$ and $\mu$ and the entropic index q, are estimated by a short comparison between theoretical predictions and observational data from cosmic ray experiments. Our findings suggest that the integration of fractional calculus and non-extensive statistics can be employed for describing the cosmic ray propagation and the anomalous diffusion observed in the heliosphere. Full article

This study addresses the environmental challenges posed by consumerism, evaluating the impact of Degradation-Resistant Organic Compounds (DROCs), such as fats and oils, on surface environments, the subsurface, groundwater, and aquifers. Climate variability has intensified the dispersion of these compounds, particularly in open landfills and poorly managed sites, making it urgent to identify affected areas to plan remediation efforts and mitigate their effects on ecosystems and human health. The objective was to analyze the dispersion of DROCs in an abandoned landfill in northwestern Mexico and develop strategies to characterize the decomposition stages of these compounds. In 2021 and 2022, a two-dimensional geoelectric tomography (GT) was conducted using a SARIS-Scintrex resistivimeter, recording variations in the apparent resistivity of subsoil. Using resistivity measures, the progressive and regressive numerical interpolation of Newton for finite differences, combined with the Lévy-type diffusion operator, classified the invasion areas into three principal ranges: high (recent invasions), intermediate (transition phase), and low (mature invasions).. These ranges indicated how pollutants migrate from the surface to the saturated zones of the aquifer. To validate the presence of fats and oils, a 24-m-deep well was drilled, revealing a positive correlation (R² = 0.863) between the areas covered by the tomograms and the detected contaminants. The results emphasize the need for improved waste management and the careful selection of disposal sites to reduce environmental degradation. The methodology proved effective and rapid, facilitating remediation planning and highlighting the importance of sustainable practices in final disposal to mitigate the impacts of DROCs, reduce greenhouse gas emissions, and protect public and environmental health. Full article

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents $\{{\gamma }_1,{\gamma }_2,\dots \}$ under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. Full article

The segmentation analysis of the Golding–Cox mRNA dataset clarifies the description of these trajectories as a Fractional Lévy Stable Motion (FLSM). The FLSM method has several important advantages. Using only a few parameters, it allows for the detection of jumps in segmented trajectories with non-Gaussian confined parts. The value of each parameter indicates the contribution of confined segments. Non-Gaussian features in mRNA trajectories are attributed to trajectory segmentation. Each segment can be in one of the following diffusion modes: free diffusion, confined motion, and immobility. When free diffusion segments alternate with confined or immobile segments, the mean square displacement of the segmented trajectory resembles subdiffusion. Confined segments have both Gaussian (normal) and non-Gaussian statistics. If random trajectories are estimated as FLSM, they can exhibit either subdiffusion or Lévy diffusion. This approach can be useful for analyzing empirical data with non-Gaussian behavior, and statistical classification of diffusion trajectories helps reveal anomalous dynamics. Full article

The paper examines the valuation and hedging of life insurance obligations in the presence of mortality risk using the local risk-minimizing hedging approach. Roughly speaking, it is assumed that the lifetime of policyholders in an insurance portfolio is modeled by a point process whose stochastic intensity is controlled by a diffusion process. The stock price process is assumed to be a regime-switching Lévy process with non-zero regime-switching drift, where the parameters are assumed to depend on the economic states. Using the Föllmer–Schweizer decomposition, the main valuation and hedging results for a conditional payment process are determined. Some specific situations have been considered in which the local risk-minimizing strategies for a stream of liability payments or a unit-linked contract are presented. Full article

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes. Full article

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate. Full article

This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in ${\mathcal{L}}^p(p\ge 2)$ sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in ${\mathcal{L}}^p(0<p<2)$ sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples. Full article

In the study of femtoscopic correlations in high-energy physics, besides Bose–Einstein correlations, one has to take final-state interactions into account. Amongst them, Coulomb interactions play a prominent role in the case of charged particles. Recent measurements have shown that in heavy-ion collisions, Bose–Einstein correlations can be best described by Lévy-type sources instead of the more common Gaussian assumption. Furthermore, three-dimensional measurements have indicated that, depending on the choice of frame, a deviation from spherical symmetry observed under the assumption of Gaussian source functions persists in the case of Lévy-type sources. To clarify such three-dimensional Lévy-type correlation measurements, it is thus important to study the effect of Coulomb interactions in the case of non-spherical Lévy sources. We calculated the Coulomb correction factor numerically in the case of such a source function for assorted kinematic domains and parameter values using the Metropolis–Hastings algorithm and compared our results with previous methods to treat Coulomb interactions in the presence of Lévy sources. Full article