Search Results (64)

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

Small, forested catchments are prototypes of terrestrial ecosystems and have been studied in several disciplines of environmental science over several decades. Time series of water and matter fluxes and nutrient concentrations from these systems exhibit a bewildering diversity of spatiotemporal patterns, indicating the intricate nature of processes acting on a large range of time scales. Nonlinear dynamics is an obvious framework to investigate catchment time series. We analyzed selected long-term data from three headwater catchments in the Bramke valley, Harz mountains, Lower Saxony in Germany at common biweekly resolution for the period 1991 to 2023. For every time series, we performed gap filling, detrending, and removal of the annual cycle using singular system analysis (SSA), and then calculated metrics based on ordinal pattern statistics: the permutation entropy, permutation complexity, and Fisher information, as well as their generalized versions (q-entropy and α-entropy). Further, the position of each variable in Tarnopolski diagrams is displayed and compared to reference stochastic processes, like fractional Brownian motion, fractional Gaussian noise, and β noise. Still another way of distinguishing deterministic chaos and structured noise, and quantifying the latter, is provided by the complexity from ordinal pattern positioned slopes (COPPS). We also constructed horizontal visibility graphs and estimated the exponent of the decay of the degree distribution. Taken together, the analyses create a characterization of the dynamics of these systems which can be scrutinized for universality, either across variables or between the three geographically very close catchments. Full article

The need to manage the risks related to the COVID-19 epidemic in health, economics, finance, and insurance became obvious after its outbreak. As a basis for the respective quantitative methods, this paper models, in a novel manner, the dynamics of an epidemic via a four-dimensional stochastic differential equation. Crucial time-dependent input parameters include the reproduction number, the average number of externally and newly infected cases, and the average number of new vaccinations. The proposed model is driven by a single Brownian motion process. When fitted to COVID-19 data, it generates the observed features. It captures widely observed fluctuations in the number of newly infected cases. The fundamental probabilistic properties of the dynamics of an epidemic can be deduced from the proposed model. These can form the basis for successfully managing an epidemic and the related economic and financial risks. As a general tool for quantitative studies, a simulation algorithm is provided. A case study illustrates the model and discusses strategies for reopening an economy during an epidemic. Full article

Our aim in this paper is to analytically compute the at-the-money second derivative of the Bachelier implied volatility curve as a function of the strike price for correlated stochastic volatility models. We also obtain an expression for the short-term limit of this second derivative in terms of the first and second Malliavin derivatives of the volatility process and the correlation parameter. Our analysis does not need the volatility to be Markovian and can be applied to the case of fractional volatility models, both with $H<1/2$ and $H>1/2.$ More precisely, we start our analysis with an adequate decomposition formula of the curvature as the curvature in the uncorrelated case (where the Brownian motions describing asset price and volatility dynamics are uncorrelated) plus a term due to the correlation. Then, we compute the curvature in the uncorrelated case via Malliavin calculus. Finally, we add the corresponding correlation correction and we take limits as the time to maturity tends to zero. The presented results can be an interesting tool in financial modeling and in the computation of the corresponding Greeks. Moreover, they allow us to obtain general formulas that can be applied to a wide class of models. Finally, they provide us with a precise interpretation of the impact of the Hurst parameter H on this curvature. Full article

In nature, closely related species often exhibit diverse characteristics, challenging simplistic line interpretations of trait evolution. For these species, the evolutionary dynamics of one trait may differ markedly from another, with some traits evolving at a slower pace and others rapidly diversifying. In light of this complexity and concerning the phenomenon of trait relationships that escape line measurement, we introduce a novel general adaptive optimal regression model, grounded on polynomial relationships. This approach seeks to capture intricate patterns in trait evolution by considering them as continuous stochastic variables along a phylogenetic tree. Using polynomial functions, the model offers a holistic and comprehensive description of the traits of the studied species, accounting for both decreasing and increasing trends over evolutionary time. We propose two sets of optimal adaptive evolutionary polynomial regression models of $k^{th}$ order, named the Ornstein–Uhlenbeck Brownian Motion Polynomial (${\mathrm{OUBMP}}_k$) model and Ornstein–Uhlenbeck Ornstein–Uhlenbeck Polynomial (${\mathrm{OUOUP}}_k$) model, respectively. Assume that the main trait value $y_t$ is a random variable of the Ornstein–Uhlenbeck (OU) process and that its optimal adaptive value ${\theta }_t^y$ has a polynomial relationship with other traits $x_t$ for statistical modeling, where $x_t$ can be a random variable of Brownian motion (BM) or OU process. As analytical representations for the likelihood of the models are not feasible, we implement an approximate Bayesian computation (ABC) technique to assess the performance through simulation. We also plan to apply models to the empirical study using the two datasets: the longevity vs. fecundity in the Mediterranean nekton group, and the trophic niche breadth vs. body mass in carnivores in a European forest region. Full article

Based on allometric theory and scaling laws, numerous mathematical models have been proposed to study ontogenetic growth patterns of animals. Although deterministic models have provided valuable insight into growth dynamics, animal growth often deviates from strict deterministic patterns due to stochastic factors such as genetic variation and environmental fluctuations. In this study, we extend a general model for ontogenetic growth proposed by West et al. to stochastic models for ontogenetic growth by incorporating stochasticity using white noise. According to data variance fitting for stochasticity, we propose two stochastic models for ontogenetic growth, one is for determinate growth and one is for indeterminate growth. To develop a universal stochastic process for ontogenetic growth across diverse species, we approximate stochastic trajectories of two stochastic models, apply random time change, and obtain a geometric Brownian motion with a multiplier of an exponential time factor. We conduct detailed mathematical analysis and numerical analysis for our stochastic models. Our stochastic models not only predict average growth well but also variations in growth within species. This stochastic framework may be extended to studies of other growth phenomena. Full article

The time irreversibility of a dynamical process refers to the phenomenon where its behaviour or statistical properties change when it is observed under a time-reversal operation, i.e., backwards in time and indicates the presence of an “arrow of time”. It is an important feature of both synthetic and real-world systems, as it represents a macroscopic property that describes the mechanisms driving the dynamics at a microscale level and that stems from non-linearities and the presence of non-conservative forces within them. While many alternatives have been proposed in recent decades to assess this feature in experimental time series, the evaluation of their performance is hindered by the lack of benchmark time series of known reversibility. To solve this problem, we here propose and evaluate the use of a geometric Brownian motion model with stochastic resetting. We specifically use synthetic time series generated with this model to evaluate eight irreversibility tests in terms of sensitivity with respect to several characteristics, including their degree of irreversibility and length. We show how tests yield at times contradictory results, including the false detection of irreversible dynamics in time-reversible systems with a frequency higher than expected by chance and how most of them detect a multi-scale irreversibility structure that is not present in the underlying data. Full article

In an AI-immersing age, scholars look for new possibilities of employing AI technology to their fields, and how to strengthen security and protect privacy is no exception. In a coverless data hiding domain, the embedding capacity of an image generally depends on the size of a chosen database. Therefore, choosing a suitable database is a critical issue in coverless data hiding. A novel coverless data hiding approach is proposed by applying deep learning models to generate texture-like cover images or code images. These code images are then used to construct steganographic images to transmit covert messages. Effective mapping tables between code images in the database and hash sequences are established during the process. The cover images generated by a two-dimensional fractional Brownian motion (2D FBM) are simply called fractional Brownian images (FBIs). The only parameter, the Hurst exponent, of the 2D FBM determines the patterns of these cover images, and the seeds of a random number generator determine the various appearances of a pattern. Through the 2D FBM, we can easily generate as many FBIs of multifarious sizes, patterns, and appearances as possible whenever and wherever. In the paper, a deep learning model is treated as a secret key selecting qualified FBIs as code images to encode corresponding hash sequences. Both different seeds and different deep learning models can pick out diverse qualified FBIs. The proposed coverless data hiding scheme is effective when the amount of secret data is limited. The experimental results show that our proposed approach is more reliable, efficient, and of higher embedding capacity, compared to other coverless data hiding methods. Full article

Online maps are of great importance in modern life, especially in commuting, traveling and urban planning. The accessibility of remote sensing (RS) images has contributed to the widespread practice of generating online maps based on RS images. The previous works leverage an idea of domain mapping to achieve end-to-end remote sensing image-to-map translation (RSMT). Although existing methods are effective and efficient for online map generation, generated online maps still suffer from ground features distortion and boundary inaccuracy to a certain extent. Recently, the emergence of diffusion models has signaled a significant advance in high-fidelity image synthesis. Based on rigorous mathematical theories, denoising diffusion models can offer controllable generation in sampling process, which are very suitable for end-to-end RSMT. Therefore, we design a novel end-to-end diffusion model to generate online maps directly from remote sensing images, called MapGen-Diff. We leverage a strategy inspired by Brownian motion to make a trade-off between the diversity and the accuracy of generation process. Meanwhile, an image compression module is proposed to map the raw images into the latent space for capturing more perception features. In order to enhance the geometric accuracy of ground features, a consistency regularization is designed, which allows the model to generate maps with clearer boundaries and colorization. Compared to several state-of-the-art methods, the proposed MapGen-Diff achieves outstanding performance, especially a $5\%$ RMSE and $7\%$ SSIM improvement on Los Angeles and Toronto datasets. The visualization results also demonstrate more accurate local details and higher quality. Full article

Composition formula is one of the most important research topics in functional analysis theory. Various relationships can be obtained using the composition formula. Since the generalized Brownian motion process used in this paper has a non-zero mean function, there are many restrictions on obtaining a composition formula for the modified analytic function space Fourier–Feynman transform. This paper contains an idea of how the composition Formula (9) below is established for the modified analytic function space Fourier–Feynman transform on function space. Using this idea, we are able to solve a problem that had never been solved before. 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

This paper adopted numerical simulation based on the MD method to research the effect of different shear rates and wax contents on wax deposition focused on crude oil. The findings indicated that under shear flow conditions, there were primarily four steps during deposition. Diffusion was the initial stage when wax diffused onto the metal surface. In the second stage, wax adsorbed onto a metal surface aligned itself parallel to the surface via Brownian motion, generating two different kinds of deposits. Subsequently, agglomerates were formed between the adsorbed deposits and the wax as a result of molecular interactions and bridging effects. Furthermore, the second and third deposited layers gradually showed peeling off and sliding under shear force. The wax deposition process was comparable for crude oil systems with varying shear rates and wax concentrations, and the deposited layer’s thickness on the metal surface was constant. The first, second, and third deposits were mainly adsorbed at 0.122 nm, 0.532 nm, and 1.004 nm away from the Fe surface, and the interaction energy between crude oil molecules and the Fe surface was mainly vdW force. The contact between Fe and wax progressively increased as the shear rate and wax content rose, promoting the wax adsorption on the metal surface and causing more of the wax to congregate in the deposited wax. The findings of the research can theoretically help a more thorough comprehension of the wax deposition. Full article

In this study, we explore backward stochastic differential equations driven by a Poisson process and an independent Brownian motion, denoted for short as BSDEJs. The generator exhibits logarithmic growth in both the state variable and the Brownian component while maintaining Lipschitz continuity with respect to the jump component. Our study rigorously establishes the existence and uniqueness of solutions within suitable functional spaces. Additionally, we relax the Lipschitz condition on the Poisson component, permitting the generator to exhibit logarithmic growth with respect to all variables. Taking a step further, we employ an exponential transformation to establish an equivalence between a solution of a BSDEJ exhibiting quadratic growth in the z-variable and a BSDEJ showing a logarithmic growth with respect to y and z. Full article

In this paper, we investigate the Green measure for a class of non-Gaussian processes in ${\mathbb{R}}^d$. These measures are associated with the family of generalized grey Brownian motions $B_{\beta ,\alpha }$, $0<\beta \le 1$, $0<\alpha \le 2$. This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability 1 for $d\alpha >2$ and $1<\alpha \le 2$. The Green measure then generalizes those measures of all these classes. Full article