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This work develops a novel class of numerical approximation algorithms for highly nonlinear stochastic differential equations. It is inspired by a stochastic approximation/optimization algorithm. The idea is the generation of random-varying truncation bounds. The algorithms are suited in case the coefficients have faster than linear growth resulting in the finite explosion time in implementing the usual Euler-Maruyama scheme, and are easier to be implemented in contrast to the existing approaches. In this paper, weak convergence and weak convergence rates of the algorithms are established using the martingale problem formulation. Some numerical examples are presented for demonstration. Finally, remarks on algorithms with additional random switching and algorithms with decreasing step sizes are given. [ABSTRACT FROM AUTHOR]

This manuscript introduces the development of the stochastic Sumudu transform theory of Itô type for stochastic calculus. We employ the stochastic integration by parts method to achieve this. The purpose of the stochastic Sumudu transform is to solve stochastic differential equations and establish a method for solving them using integral transforms. Furthermore, we derive the Sumudu transforms of commonly used functions in stochastic differential equations. These findings will contribute to the enhancement of literature on stochastic differential equations and have practical applications in fields such as applied mathematics and finance. Additionally, we provide several examples to demonstrate the validity of our work. [ABSTRACT FROM AUTHOR]

This study aimed to examine changes in the number of live and dying trees in central Lithuanian forests over time. Results were obtained using stochastic differential equations combined with the normal copula function. The examination of each tree's individual size variables (height and diameter) showed that the mean values of dead or dying trees' size variables had significantly lower trajectories that were particularly pronounced in mature stands. According to the data set under examination, the tree mortality rate gradually declined with age, reaching approximately 7% after 10 years. Birch trees 60–70 years old were the first species to reach the 1% mortality rate, followed by spruce trees 70–80 years old and pine trees 80–90 years old. The Maple symbolic algebra system was used to implement all results. [ABSTRACT FROM AUTHOR]

The reconstruction of stochastic evolution equations from time-series data in terms of the Langevin equation and the corresponding Fokker–Planck equation is often challenged by the inevitably finite temporal sampling of time-series data. With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, one can use a finite sampling τ expansion of the Kramers–Moyal (KM) conditional moments. Depending on the number of terms used from such expansion one obtains different orders of approximation of the KM coefficients. In this paper, we derive and employ the second-order approximation ∼τ2 of the KM conditional moments for the Langevin dynamics and demonstrate that its use leads to the modification of the diffusion coefficient. To demonstrate the substantial potential for practical applications, we utilize data-driven techniques to estimate fluctuations in synthetic time series generated from both linear and nonlinear diffusive processes. Furthermore, we show that when Langevin dynamics is specifically used to model asset prices (here gold price), the use of the second-order approximation of the KM coefficients leads to more accurate prices simulation and prediction that have not been observed in similar work by others. [ABSTRACT FROM AUTHOR]

In the current work, we develop and analyse a prey–predator model as a semi-Kolmogorov population model in which the predator has an indirect effect on the prey. The functional response of the model is investigated as Holling type (II). We construct a stochastic environment because of the parameter's random essence to study the influence of environmental fluctuations on this model and introduce a stochastic version of the prey–predator model. Then we present a dynamical analysis of solutions, including existence, uniqueness, positivity, stochastic boundedness, and stochastic extinction of all prey. Finally, some numerical simulations are carried out to validate our theoretical findings and confirm the efficiency and adaptation of our stochastic model. [ABSTRACT FROM AUTHOR]

In this work, we study the qualitative properties of a simple mathematical model that can be applied to the reversal of antimicrobial resistance. In particular, we analyze the model from three perspectives: ordinary differential equations (ODEs), stochastic differential equations (SDEs) driven by Brownian motion, and fractional differential equations (FDEs) with Caputo temporal derivatives. Finally, we address the case of Escherichia coli exposed to colistin using parameters from the literature in order to assess the validity of the qualitative properties of the model.

In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $ M : \mathbb{R}^n \rightarrow \mathbb{R}^n $ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process $ X(t) $ to a zero of $ M $ and of $ \|M(X(t))\| $ to $ 0 $ as $ t \rightarrow +\infty $. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for $ \|M(X(t))\|^2 $ and $ \langle M(X(t), X(t)-x^*\rangle $, where $ x^* $ is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a type of Pareto game of stochastic differential equation with terminal state constraint. Firstly, we transform equivalently a nonlinear Pareto game problem with convex control space and terminal state constraint into a constrained stochastic optimal control problem. By virtue of duality theory and stochastic maximum principle, a necessary condition for Pareto efficient strategy is established. With some convex assumptions, we also give a sufficient condition for Pareto efficient strategy. Secondly, we consider a linear-quadratic Pareto game with terminal state constraint, and a feedback representation for Pareto efficient strategy is derived. Finally, as an application, we solve a government debt stabilization problem and give some numerical results. [ABSTRACT FROM AUTHOR]

Using the ordinary differential equation of motion it is possible to determine the position in time of a mass that moves because it is disturbed by some deterministic action. For this work it was proposed to model a mass supporting a random disturbance. To do this, it was required to model Brownian motion since it efficiently represents the randomness of the phenomenon. Using the fundamentals of Functional Analysis, Probability Theory and Stochastic Processes, a stochastic differential equation of motion was obtained. In order to extract solutions from this equation, the Euler-Maruyama method was used, which was implemented computationally. The results obtained showed that the use of a non-deterministic version to model movement generates satisfactory results and of interest to science.

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.

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]

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]

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]

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]

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. [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]

The paper presents a survey of the results obtained by the members of the Department of Mathematical Statistics in the field of analytic and asymptotic properties of mixture probability models. Much attention is paid to the representation of some widely applied absolutely continuous probability distributions (gamma, Weibull, Student, Snedecor–Fisher, Mittag-Leffler, Burr, etc.) as mixtures of distributions possessing maximum differential entropy (normal and exponential). Some useful discrete distributions that are representable as mixed Poisson distributions are discussed as well. Examples are presented of limit theorems for statistics constructed from samples with random sizes in which the distributions mentioned above are limit laws. Also, the estimates of convergence rate in these theorems are presented. Some problems related to the application of methods of intellectual analysis of big arrays of dynamically accumulated data based on mixture probability models are discussed. [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 paper, we present a new precipitation model based on a multi-factor Ornstein-Uhlenbeck approach of pure-jump type. In this setup, we derive a representation for the related precipitation swap price process and infer its risk-neutral time dynamics. We further deduce a pricing formula for European options written on the precipitation swap and obtain the minimal variance hedging portfolio in the underlying weather market. In the second part of the paper, we provide a precipitation swap price representation under future information modeled by an initially enlarged filtration. We finally derive a formula for the associated information premium and investigate minimal variance hedging of precipitation derivatives under future information. [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]

Parameter estimation of stochastic differential equation has recently been discussed by many authors. The aim of this paper is to study the rates of convergence of approximate maximum likelihood estimator. More precisely, for Markovian switching diffusions, we first show the convergence rates of the continuous maximum likelihood estimator under the Lipschitz conditions. Then we also discuss the probabilistic bounds on |θn;T - θT| under the non-Lipschitz conditions. [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]

We discuss the design of an invariant measure-preserving transformation for the numerical treatment of Langevin dynamics based on a rescaling of time, with the goal of sampling from an invariant measure. Given an appropriate monitor function which characterizes the numerical difficulty of the problem as a function of the state of the system, this method allows stepsizes to be reduced only when necessary, facilitating efficient recovery of long-time behavior. We study both overdamped and underdamped Langevin dynamics. We investigate how an appropriate correction term that ensures preservation of the invariant measure should be incorporated into a numerical splitting scheme. Finally, we demonstrate the use of the technique on several model systems, including a Bayesian sampling problem with a steep prior. [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]

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]

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 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]

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]

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]

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]

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]

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]

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]