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Given a separable and real Hilbert space H , we consider the following stochastic differential equation (SDE) on H : d X t = − X t d t + b (X t) d t + d Z t , where Z : = (Z t) t ≥ 0 is a cylindrical pure jump Lévy process on H which may be degenerate in the sense that the support of Z is contained in a finite dimensional space. When the nonlinear drift term b (x) is contractive with respect to some proper modified norm of H for large distances, we obtain explicit exponential contraction rates of the SDE above in terms of Wasserstein distance under mild assumptions on the Lévy process Z. The approach is based on the refined basic coupling of Lévy noises, and it also works well when the so-called Lyapunov condition is satisfied. [ABSTRACT FROM AUTHOR]

A method for calculating the characteristics of stochastic differential equations using semiclassical approximation is proposed. For a stochastic differential equation arising in the study of the pure birth process (Yule process) and the Cox Ingersoll Ross (CIR) model, an analysis of the accuracy of the semiclassical approximation was carried out. This analysis is based on a comparison of approximate values with exact values for the mathematical expectation of a solution to the equation. [ABSTRACT FROM AUTHOR]

The broaching that follows the surf-riding is a dangerous phenomenon that can lead to the capsizing of a vessel due to its violent yaw motion. Most of the previous studies on surf-riding phenomena in irregular waves have been conducted by replacing irregular waves with regular waves. In contrast, this study provides suggestions on how to directly calculate nonlinear surge motion in irregular seas. In this study, the statistical aspects of the surf-riding phenomenon are first presented. Then, under several approximations, we show how to calculate the probability density function theoretically. Although the results obtained are based on strong approximations, it is found that the nonlinear surge oscillations in irregular following seas can be explained from a qualitative point of view. [ABSTRACT FROM AUTHOR]

Recently, a considerable number of research and development projects have focused on automatic vessels. A highly realistic simulator is needed to validate control algorithms for autonomous vessels. For instance, when considering the automatic berthing/unberthing of a vessel, the effect of wind in such low-speed operations cannot be ignored because of the low rudder performance during slow harbor maneuvers. Therefore, a simulator used to validate an automatic berthing/unberthing control algorithm should be able to reproduce the time histories of wind speed and wind direction realistically. Therefore, in our first report on this topic, to obtain the wind speed distribution, we proposed a simple algorithm to generate the time series and distribution of wind speed only from the mean wind speed. However, for wind direction, the spectral distribution could not be determined based on our literature surveys, and hence, a simple method for estimating the coefficients of the stochastic differential equation (SDE) could not be proposed. In this study, we propose a new methodology for generating the time history of wind direction based on the results of Kuwajima et al.'s work. They proposed a regression equation of the standard deviation of wind direction variation for the mean wind speed. In this study, we assumed that the wind direction distribution can be represented by a linear filter as in our previous paper, and its coefficients are derived from Kuwajima's proposed equation. Then, as in the previous report, the time series of wind speed and wind direction can be calculated easily by analytically solving the one-dimensional SDE. The joint probability density functions of wind speed and wind direction obtained by computing them independently agree well with the measurement results. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic properties of a kernel-type nonparametric estimator of the linear multiplier in models governed by a stochastic differential equation driven by a general Gaussian process. [ABSTRACT FROM AUTHOR]

This work discusses the time-varying formation (TVF) tracking control problem of high-order multi-agent systems (MASs) with multiple leaders and multiplicative measurement noise. With the help of Lyapunov function tools and stochastic analysis methods, the TVF tracking protocol with multiple leaders and multiplicative noise is developed based on the relative state measurements, where followers are driven to realize the target TVF while tracking the convex combination formed by multiple leaders. Then, the TVF tracking problem is converted into the mean square asymptotic stability problem of a stochastic differential equation (SDE); sufficient conditions related to the control gains are given by stabilizing the corresponding stochastic system. Moreover, a TVF tracking algorithm is presented to outline the steps of protocol design. Finally, the theoretical results are illustrated in terms of simulation examples. [ABSTRACT FROM AUTHOR]

The main objective of this article is to study the effectiveness of wearing masks and implementing active screening and testing to contain the spread of COVID-19 during stages when the disease has already started to spread. This work innovates by focusing on the basic measures of mask use and active screening and testing, which are frequently overlooked in the literature in favor of more expensive interventions like travel bans and vaccinations. We study their effectiveness and determine the optimal way to implement them. To achieve this goal, we develop a stochastic SEIR model that includes four populations: susceptible, exposed, infected, and recovered individuals. We integrate two control functions into the model to represent mask-wearing and active screening and testing interventions. Using an adapted formulation of Pontryagin's maximum principle suitable for the stochastic context, we aim to find the optimal approach to implementing these controls. We combine the Forward-Backward Sweep Method (FBSM) with a special Runge–Kutta scheme for stochastic differential equations to solve the optimality system for our stochastic optimal control problem. Our study demonstrates that even when the disease has begun to spread, using both mask-wearing and active screening and testing can lead to significant reductions in the numbers of exposed and infected individuals, offering a cost-effective strategy for countries with limited resources. [ABSTRACT FROM AUTHOR]

In this research paper, a mathematical concept known as the ω-periodic process is discussed, and a new type of processes called the doubly measure pseudo -asymptotically ω-periodic in distribution process is set forward. Additionally, the properties of these processes are identified and invested to tackle the solution of a stochastic differential equation guided by Brownian motion. The basic objective of the current work resides in corroborating the existence and uniqueness of the solution which is doubly measure pseudo -asymptotically ω-periodic in distribution. [ABSTRACT FROM AUTHOR]

The accurate knowledge of cosmic ion fluxes is essential for fundamental physics, deep space missions, and exploration activities in the solar system. In the HelMod-4 model the Parker transport equation is solved using a Monte Carlo approach to evaluate the solar modulation effect on local interstellar spectra of Galactic Cosmic Rays (GCRs). This work presents the latest updates to the HelMod-4 model parameters, focusing on the descending phase of solar cycle 24. The updates are motivated by the latest high-precision measurements from the AMS-02 detector which revealed for the first time with high accuracy the features of GCR fluxes' evolution during a period of positive interplanetary magnetic field polarity. Furthermore, we present HelMod-4/CUDA , a GPU-accelerated approach for solving the Parker equation in the heliosphere using a stochastic differential equation method. The code is an evolution of the HelMod-4 code, porting the algorithm to GPU architecture using the CUDA programming language. This approach achieves significant speedup compared to a CPU implementation. The HelMod-4/CUDA code has been validated by comparing its results with the most precise and updated experimental GCR spectra observed during high and low solar activity periods, both in the inner and outer heliosphere, at the Earth location, and outside the ecliptic plane. The comparison shows that HelMod-4 and HelMod-4/CUDA can be equivalently used to provide solar-modulated spectra with a similar degree of accuracy in reproducing observed data. [ABSTRACT FROM AUTHOR]

The threshold Ornstein–Uhlenbeck process is a stochastic process that satisfies a stochastic differential equation with a drift term modeled as a piecewise linear function and a diffusion term characterized by a positive constant. This paper addresses the challenge of determining both the number and values of thresholds based on the continuously observed process. We present three testing algorithms aimed at determining the unknown number and values of thresholds, in conjunction with least squares estimators for drift parameters. The limiting distribution of the proposed test statistic is derived. Additionally, we employ sequential and global methods to determine the values of thresholds, and prove their weak convergence. Monte Carlo simulation results are provided to illustrate and support our theoretical findings. We utilize the model to estimate the term structure of US treasury rates and currency foreign exchange rates. [ABSTRACT FROM AUTHOR]

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated. [ABSTRACT FROM AUTHOR]

This paper proves the qualitative properties like existence, uniqueness and Ulam-Hyers-Rassias stability for a solution of a nonlinear impulsive Hilfer fractional stochastic differential equation with non-instantaneous impulses. The Banach fixed point theorem and Cauchy inequality are utilized for obtaining our results. The Ulam-Hyers-Rassias stability is presented under specific assumptions. To verify the theoretical results an example is presented at the end. [ABSTRACT FROM AUTHOR]

In this study, an intelligent computing framework is presented to explore the influence of unpredictable environmental factors on the spread of infections. A stochastic non-linear SIS epidemic model (SISEM) is examined that incorporates a non-linear incidence rate, and impact of random fluctuations on the disease transmission. The knacks of artificial neural networks Levenberg–Marquardt (ANNLMA) technique are exploited to illustrate the behavioral complexities of stochastic non-linear SISEM, which explores the dynamics of disease transmission in a stochastic environment and provides insights into the interactions between susceptible and infectious populations. The Milstein approach of order 1 generates a dataset for the proposed ANNLMA technique by varying integrated parameters, such as the rates of recovery, death, disease transmission, noise standard deviation and the entire population, for various SISEM scenarios. The referenced dataset is randomly divided into test, validation and train sets for the weight learning of the proposed ANNLMA. The viability of the ANNLMA is measured using intensive simulation-based investigations through analysis of mean squared errors, histograms, regression and autocorrelation studies. The investigations highlighted that the ANNLMA results closely match the reference results, illustrating convergence within the 10−2 to 10−7 range, indicating the validity of the proposed methodology. [ABSTRACT FROM AUTHOR]

The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker–Planck equation using the standard toolbox of Riemann geometry. The covariant form of the adapted Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free. [ABSTRACT FROM AUTHOR]

Marcus stochastic delay differential equations are often used to model stochastic dynamical systems with memory in science and engineering. It is challenging to study the existence, uniqueness, and probability density of Marcus stochastic delay differential equations, due to the fact that the delays cause very complicated correction terms. In this paper, we identify Marcus stochastic delay differential equations with some Marcus stochastic differential equations without delays but subject to extra constraints. This helps us to obtain the following two main results: (i) we establish a sufficient condition for the existence and uniqueness of the solution to the Marcus delay differential equations; and (ii) we establish a representation formula for the probability density of the Marcus stochastic delay differential equations. In the representation formula, the probability density for Marcus stochastic differential equations with memory is analytically expressed in terms of probability density for the corresponding Marcus stochastic differential equations without memory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1) . By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present two important corollaries on the non-confluence property by the Poisson equation and M-matrix, respectively, which can verify the non-confluence property more effectively than the general condition. Finally, we provide an example to illustrate the practical usefulness of our theoretical results. [ABSTRACT FROM AUTHOR]

Given global Lipschitz continuity and differentiability of high enough order on the coefficients in Itô's equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of order one for numerical approximations are known. In this work and against the counterexamples of Hairer et al. (Ann Probab 43(2):468–527, https://doi.org/10.1214/13-AOP838, 2015), the drift and diffusion coefficients having Lipschitz constants that are o (log V) and o (log V) respectively for a function V satisfying (∂ t + L) V ≤ C V is shown to be a generalizing condition in place of global Lipschitz continuity for the above. [ABSTRACT FROM AUTHOR]

The problem of thermal stability of a cylindrical sample with nonlinear heat generation placed in a medium with the ambient temperature random walk was studied. The behavior of this system was examined depending on the parameters of the problem (heat generation intensity, random walk variance). A numerical algorithm based on averaging multiple random trajectories of the ambient temperature was proposed. A numerical method was developed for solving the heat transfer problem with the heat source and stochastic boundary which combines both explicit and implicit schemes for linearized transfer equations and the Euler–Maruyama method. The distributions of ignition characteristics and their moments were obtained. Their dependencies on the parameters of the problem were investigated.

Aiming at the problem of high failure rate and inconvenient maintenance of wind turbine gearboxes, this paper establishes a stochastic differential equation model that can be used to fit the change of gearbox oil temperature and adopts an iterative computational method and Markov-based modified optimization to fit the prediction sequence in order to realize the accurate prediction of gearbox oil temperature. The model divides the oil temperature change of the gearbox into two parts, internal aging and external random perturbation, adopts the approximation theorem to establish the internal aging model, and uses Brownian motion to simulate the external random perturbation. The model parameters were calculated by the Newton–Raphson iterative method based on the gearbox oil temperature monitoring data. Iterative calculations and Markov-based corrections were performed on the model prediction data. The gearbox oil temperature variations were simulated in MATLAB, and the fitting and testing errors were calculated before and after the iterations. By comparing the fitting and testing errors with the ordinary differential equations and the stochastic differential equations before iteration, the iterated model can better reflect the gear oil temperature trend and predict the oil temperature at a specific time. The accuracy of the iterated model in terms of fitting and prediction is important for the development of preventive maintenance. [ABSTRACT FROM AUTHOR]

The goals of this paper are to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. For a class of anticipating linear stochastic differential equations, we prove the existence and uniqueness of solutions using two approaches: (1) Ayed–Kuo differential formula using an ansatz, and (2) a braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin–Wentzell type large deviations result for the solution of such equations. In addition, we prove large deviation results for small noise where the initial conditions are random. [ABSTRACT FROM AUTHOR]

We discuss nonparametric estimation of the trend coefficient in models governed by a stochastic differential equation driven by a Gaussian process with small noise. [ABSTRACT FROM AUTHOR]

The growing online retail market has led to the prevalence of multichannel retailing. Meanwhile, retailers are increasingly combining multichannel retailing with a multibranding strategy. Although this can further increase the retailer's sales, it brings new advertising challenges. Multibrand, multichannel retailers usually launch advertising campaigns for different brands on multiple media. Thus, their advertising efforts fall into a set of brand-media units. Each unit's advertising can affect the sales of all brands on all channels. To maximize the advertising returns, a retailer needs to coordinate the advertising expenditures on different units. We develop a stochastic differential equation model that helps the retailer estimate the impact of multimedia advertising on the sales of different brands on different channels. Afterward, we formulate the advertising optimization problems under four coordination strategies: noncoordination, brand coordination, media coordination, and global coordination. By solving the problem for each strategy, the retailers can obtain the optimal expenditure for each unit under that strategy. Finally, we compare the retailer's profits under four strategies. Interestingly, we find that brand or media coordination may result in a profit lower than noncoordination. Our findings provide insights regarding the selection of coordination strategies for multibrand, multichannel retailers with multimedia advertising campaigns. The growing online retail market has led to the prevalence of multichannel retailing. Meanwhile, retailers are increasingly combining multichannel retailing with a multibranding strategy. Although this can further increase the retailer's sales, it brings new advertising challenges. Multibrand, multichannel retailers usually launch advertising campaigns for different brands on multiple media. Thus, the retailer's advertising efforts fall into a set of brand-media units. Each unit's advertising efforts can affect the sales of all brands on all channels. Therefore, retailers need to coordinate the advertising efforts of different units to maximize advertising efficiency in propelling sales. So far, the optimization problem of multibrand, multimedia advertising has not been analyzed in the literature, and our study aims to bridge this gap. We develop a stochastic differential equation model to estimate the impact of multimedia advertising on sales in a multibrand, multichannel context. Using the data from a jewelry retailer in the United States, we show that our model is effective in predicting future sales driven by advertising. Afterward, we formulate the advertising optimization problems under four coordination strategies: (i) noncoordination, (ii) brand coordination, (iii) media coordination, and (iv) global coordination. By solving the problem for each strategy, the retailers can obtain the optimal expenditure for each unit under that strategy. Finally, we compare the retailer's profits under four strategies. Interestingly, we find that brand or media coordination may result in a profit lower than noncoordination. Our findings provide insights regarding the selection of coordination strategies for multibrand, multichannel retailers with multimedia advertising campaigns, especially when they cannot do global coordination. History: Ram Gopal, Senior Editor; Huaxia Rui, Associate Editor. Supplemental Material: The e-companion is available at https://doi.org/10.1287/isre.2022.0283. [ABSTRACT FROM AUTHOR]

Some new techniques are employed to release significantly the requirements on the step size of the truncated Milstein method, which was originally developed in Guo et al. [The truncated Milstein method for stochastic differential equations with commutative noise, J. Comput. Appl. Math. 338 (2018) 298–310]. The almost-sure stability of the method is also investigated. Numerical simulations are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We prove the Hölder–Dini regularity for a class of fractional Fokker–Planck–Kolmogorov equations. Then, by applying the Itô–Tanaka trick, we obtain the unique strong solvability of time inhomogeneous stochastic differential equations with drift coefficients in L t ∞ ( b , d β , ρ) for β = 1 − α / 2 and driven by α -stable processes. [ABSTRACT FROM AUTHOR]

Cai, Guo and Mao (2024) recently constructed a structure preserving numerical method for the n-dimensional stochastic Lotka–Volterra model and showed its strong convergence. Nevertheless, people are also concerned with whether the numerical methods are capable of reflecting the true dynamical behaviors. In this work, we will further show that this numerical method is able to reproduce the extinction of the population model under the same parametric condition. The main results are finally illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order (α) have minimal influence on the system when the noise intensity (σ) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain. [ABSTRACT FROM AUTHOR]

This paper proposes a bootstrap testing approach based on an approximate maximum likelihood method to discern whether a diffusion process is linear or whether there are threshold effects in the drift, the diffusion term or in both. It complements an alternative method based on the least-squares estimator which focuses on threshold effects in the drift. Monte Carlo simulations illustrate that the proposed testing approach is able to detect the source of the non-linearity. Two empirical applications show the importance of modeling threshold effects in the diffusion instead of the drift. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of a class of stochastic differential equations with tempered φ-Caputo fractional derivative (φ-CFD) and Lévy noise. We propose comprehensive mathematical techniques to address the existence, uniqueness and stability of solution to this equation. For existence and uniqueness, the Picard–Lindelof successive approximation technique is used analyze the results. Also, We use Mittag-Leffler (M-L) function to investigate the stability of the solution. This research applies the broad understanding of stochastic processes and fractional differential equations, as well as known results, to the analysis of systems with tempered φ-CFD. These equations capture complex phenomena in the field of financial assets, making their investigation on the stock prices particularly valuable.

This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy. [ABSTRACT FROM AUTHOR]

We study the unique solvability of the Cauchy problem for a mixed-type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst exponents . We prove a theorem on the existence and uniqueness of strong solutions of these stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]

Numerous accidents caused by parametric rolling have been reported on container ships and pure car carriers (PCCs). Considering this dangerous phenomenon, the parametric rolling in irregular seas is examined in this paper based on the stability of the systems origin, which corresponds to the upright condition of the vessel. It provides a novel theoretical explanation of the instability mechanism for two cases: white-noise parametric excitation and colored-noise parametric excitation. Moreover, the authors confirm the usefulness of the previously provided formulae by Roberts and Dostal by means of numerical examples. [ABSTRACT FROM AUTHOR]

This paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic differential equations with fixed delays and obtain two conditions to guarantee that the zero solution is globally exponentially stable in mean square by using Gronwall inequality and matric theory, respectively. The results are new and interesting. Some examples are given to illustrate the correctness and effectiveness of our results. [ABSTRACT FROM AUTHOR]

A way of intelligently predicting random processes is described, based on more complete use of information about statistical patterns of the evolution of an observed process. At the stage of training the predictive algorithm, the feature space is enriched with the parameters of mixed probabilistic models that allow the construction and reconstruction of the coefficients of a stochastic differential equation describing the given random process. The use of additional statistical information imposes additional conditions on the search area and therefore narrows the set of considered options. Learning is thus targeted by excluding impossible or unlikely options in advance, making it more effective and forecasts more accurate. [ABSTRACT FROM AUTHOR]

Contrast resolution is an important index for evaluating the signal detectability of computed tomographic (CT) images. Recently, various noise reduction algorithms, such as iterative reconstruction (IR) and deep learning reconstruction (DLR), have been proposed to reduce the image noise in CT images. However, these algorithms cause changes in the image noise texture and blurred image signals in CT images. Furthermore, the contrast-to-noise ratio (CNR) cannot be accurately evaluated in CT images reconstructed using noise reduction methods. Therefore, in this study, we devised a new method, namely, "effective CNR analysis," for evaluating the contrast resolution of CT images. We verified whether the proposed algorithm could evaluate the effective contrast resolution based on the signal detectability of CT images. The findings showed that the effective CNR values obtained using the proposed method correlated well with the subjective visual impressions of CT images. To investigate whether signal detectability was appropriately evaluated using effective CNR analysis, the conventional CNR analysis method was compared with the proposed method. The CNRs of the IR and DLR images calculated using conventional CNR analysis were 13.2 and 10.7, respectively. By contrast, those calculated using the effective CNR analysis were estimated to be 0.7 and 1.1, respectively. Considering that the signal visibility of DLR images was superior to that of IR images, our proposed effective CNR analysis was shown to be appropriate for evaluating the contrast resolution of CT images. [ABSTRACT FROM AUTHOR]

We introduce a novel stochastically correlated two-factor (i.e., bivariate) diffusion process under the square-root format, for which we analytically obtain the corresponding solutions for the conditional moment-generating functions and conditional characteristic functions. Such solutions recover verbatim those of the uncorrelated case which encompasses a range of processes similar to those produced by a bivariate square-root process in which entries are correlated in the standard way, that is, via a constant correlation coefficient. Note that closed-form solutions for the conditional characteristic and moment-generating functions are not available for the latter. We focus on the financial scenario of obtaining closed-form expressions for the exact price of a zero-coupon bond and Asian option prices using a Fourier cosine series method. [ABSTRACT FROM AUTHOR]

This article introduces a novel approach to forecasting gold prices over an extended period by leveraging a sophisticated stochastic process. Departing from traditional models, our proposed framework accommodates the non-Gaussian and non-homogeneous nature of gold market dynamics. Rooted in the α-stable distribution, our model captures time-dependent characteristics and exhibits flexibility in handling the distinctive features observed in real gold prices. Building upon prior research, we present a comprehensive methodology for estimating time-dependent parameters and validate its efficacy through simulations. The results affirm the universality of our stochastic model, showcasing its applicability for accurate and robust long-term predictions in gold prices. [ABSTRACT FROM AUTHOR]

Accurate prediction of photovoltaic (PV) power generation is the key to daily dispatch management and safe and stable grid operation. In order to improve the accuracy of the prediction, a finite iterative PV power prediction model with long range dependence (LRD) characteristics was developed using fractional Lévy stable motion (fLsm) and applied to a real dataset collected in the DKASC photovoltaic system in Alice Springs, Australia. The LRD prediction model considers the influence of current and past trends in the stochastic series on the future trends. Firstly, the calculation of the maximum steps prediction was introduced based on the maximum Lyapunov. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the order stochastic differential equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The PV power forecasting model with the LRD characteristics was obtained by discretization of FSDE. By comparing statistical performance indicators such as root max error, mean square error with Conv-LSTM, BiLSTM, and GA-LSTM models, the performance of the proposed fLsm model has been demonstrated. The proposed methods can provide better theoretical support for the stable and safe operation of PV grid connection. They have high reference value for grid dispatching department. [ABSTRACT FROM AUTHOR]