Search

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar–Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar–Euler method has much better stability properties than our recently designed composite Patankar–Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar–Euler methods. Numerical results suggest that the proposed balanced implicit Patankar–Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems. [ABSTRACT FROM AUTHOR]

The temperature dependence of the nuclear free induction decay in the presence of a magnetic-field gradient was found to exhibit motional narrowing in gases upon heating, a behavior that is opposite to that observed in liquids. This has led to the revision of the theoretical framework to include a more detailed description of particle trajectories since decoherence mechanisms depend on histories. In the case of free diffusion and single components, the new model yields the correct temperature trends. The inclusion of boundaries in the current formalism is not straightforward. We present a hybrid SDE-MD (stochastic differential equation - molecular dynamics) approach whereby MD is used to compute an effective viscosity and the latter is fed to the SDE to predict the line shape. The theory is in agreement with the experiments. This two-scale approach, which bridges the gap between short (molecular collisions) and long (nuclear induction) timescales, paves the way for the modeling of complex environments with boundaries, mixtures of chemical species, and intermolecular potentials. [ABSTRACT FROM AUTHOR]

The impact of micromixing and scalar dissipation rate (SDR) fluctuations on the evolution of soot particle size distribution (PSD) dynamics in non-premixed turbulent flames is investigated within the context of Large Eddy Simulation (LES). To accomplish this, a stochastic approach based on a physicochemical sectional soot model, spatially homogeneous Conditional Moment Closure (CMC) with first-order moment closure, and a stochastic differential equation for the SDR are employed, mimicking the function of an LES sub-grid scale (SGS) combustion model. After linking the magnitude of SDR fluctuations to the LES filter size, it was found that the standard deviation of observables, such as the soot volume fraction and soot number density, decreases rapidly with increasing LES filter size, which can significantly influence the intermittency of soot reproduced by models. Mean conditional expectations are also affected, indicating a closure problem, with the mean soot volume fraction, in particular, showing a decrease of up to 20% in the case considered compared to a direct numerical simulation resolution even when the Pope criterion is satisfied. The influence of SDR fluctuations on the dynamic behaviour of the soot PSD is further discussed under different working assumptions for particle transport, and the role of differential diffusion in soot oxidation- and formation-dominant regions is particularly highlighted. The findings underline the shortcomings of LES-CMC and similar mixture-fraction-based combustion models in capturing the dynamic evolution of soot at the SGS, which has significant implications for modelling the emissions of practical combustion devices. However, this study also offers a framework for better evaluating the effectiveness of these models using numerical experimentation and a stochastic model for the turbulent fluctuations of the micromixing rate. [ABSTRACT FROM AUTHOR]

The time-resolved scRNA-seq (tscRNA-seq) provides the possibility to infer physically meaningful kinetic parameters, e.g., the transcription, splicing or RNA degradation rate constants with correct magnitudes, and RNA velocities by incorporating temporal information. Previous approaches utilizing the deterministic dynamics and steady-state assumption on gene expression states are insufficient to achieve favorable results for the data involving transient process. We present a dynamical approach, Storm (Stochastic models of RNA metabolic-labeling), to overcome these limitations by solving stochastic differential equations of gene expression dynamics. The derivation reveals that the new mRNA sequencing data obeys different types of cell-specific Poisson distributions when jointly considering both biological and cell-specific technical noise. Storm deals with measured counts data directly and extends the RNA velocity methodology based on metabolic labeling scRNA-seq data to transient stochastic systems. Furthermore, we relax the constant parameter assumption over genes/cells to obtain gene-cell-specific transcription/splicing rates and gene-specific degradation rates, thus revealing time-dependent and cell-state-specific transcriptional regulations. Storm will facilitate the study of the statistical properties of tscRNA-seq data, eventually advancing our understanding of the dynamic transcription regulation during development and disease. Author summary: Intricate regulation of RNA biogenesis, such as RNA transcription, splicing and degradation, plays a critical role in most biological processes. Previous approaches have leveraged a deterministic model of spliced and unspliced RNAs to estimate kinetic parameters and quantify RNA velocity, the rate of changes in gene expression states in single cells. However, accurately estimating meaningful kinetic parameters and RNA velocity is hindered by biased capture of unspliced RNA and absence of temporal information in conventional scRNA-seq. Significant advances have been made in better measuring RNA kinetics with metabolic labeling enabled scRNA-seq, however, computational tools to analyze them lag far behind. Prior work of Dynamo provides one of the early solutions to properly model RNA metabolic labeling data, but its method still largely uses a deterministic model that only utilizes part of extreme cells and is unable to analyze datasets with significant transient dynamics. To address these challenges, we developed Storm that explicitly models transient stochastic RNA dynamics. Importantly, Storm models RNA kinetics with stochastic differential equations that explicitly account for biological and cell-specific noises. Storm is generally applicable to many metabolic labeling scRNA-seq datasets and we demonstrate excellent performance of Storm in fitting the data to capture the transient dynamics under various noise models. [ABSTRACT FROM AUTHOR]

The aim of this work is to show the numerical approximation of random periodic solutions for path‐dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous‐time Euler–Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path‐dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean‐square sense of a numerical approximation to path‐dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path‐dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure. [ABSTRACT FROM AUTHOR]

A feedback control method is proposed to asymptotically stabilize the chain-like multi-degree-of-freedom (MDOF) nonlinear structural system under purely parametric Gaussian white noises. First, the motion equation of a chain-like nonlinear structural system with n coupled elements and an undetermined control law is derived. It is reduced to a one-dimensional partially averaged Itô stochastic differential equation of the controlled Hamiltonian by applying the stochastic averaging method, which bypasses the challenge of calculating multiple Lyapunov exponents. Next, based on the dynamical programming principle, the dynamical programming equation of the averaged system with an undetermined cost function is established and addressed to obtain the optimal control law. Then, the Lyapunov exponent of the completely averaged Itô equation is derived. For the given requirement of stabilizing the system, the Lyapunov exponent is entirely fixed and used to analyze the stability of the originally chain-like MDOF nonlinear structural system. A controlled chain-like 8-DOF nonlinear structural system is taken as an example to illustrate the application and effectiveness of the proposed procedure and the effect of control forces on stabilizing the system. [ABSTRACT FROM AUTHOR]

Intrinsic thermal fluctuations within a real solid challenge the rigid body assumption that is central to Euler’s equations for the motion of a free body. Recently, we have introduced a dissipative and stochastic version of Euler’s equations in a thermodynamically consistent way (European Journal of Mechanics – A/Solids 103, 105,184 (2024)). This framework describes the evolution of both orientation and shape of a free body, incorporating internal thermal fluctuations and their concomitant dissipative mechanisms. In the present work, we demonstrate that, in the absence of angular momentum, the theory predicts that the principal axes unit vectors of a body undergo an anisotropic Brownian motion on the unit sphere, with the anisotropy arising from the body’s varying moments of inertia. The resulting equilibrium time correlation function of the principal eigenvectors decays exponentially. This theoretical prediction is confirmed in molecular dynamics simulations of small bodies. The comparison of theory and equilibrium MD simulations allow us to measure the orientational diffusion tensor. We then use this information in the Stochastic Dissipative Euler’s Equations, to describe a non-equilibrium situation of a body spinning around the unstable intermediate axis. The agreement between theory and simulations is excellent, offering a validation of the theoretical framework. [ABSTRACT FROM AUTHOR]

The rate at which aerosols spread from a point source injection, such as from a ship or other stationary pollution source, is critical for accurately representing subgrid plume spreading in a climate model. Such climate model results will guide future decisions regarding the feasibility and application of large-scale intentional marine cloud brightening (MCB). Prior modeling studies have shown that the rate at which ship plumes spread may be strongly dependent on meteorological conditions, such as precipitating versus non-precipitating boundary layers and shear. In this study, we apply a Lagrangian particle model (PM-ABL v1.0), governed by a Langevin stochastic differential equation, to create a simplified framework for predicting the rate of spreading from a ship-injected aerosol plume in sheared, precipitating, and non-precipitating boundary layers. The velocity and position of each stochastic particle is predicted with the acceleration of each particle being driven by the turbulent kinetic energy, dissipation rate, momentum variance, and mean wind. These inputs to the stochastic particle velocity equation are derived from high-fidelity large-eddy simulations (LES) equipped with a prognostic aerosol–cloud microphysics scheme (UW-SAM) to simulate an aerosol injection from a ship into a cloud-topped marine boundary layer. The resulting spreading rate from the reduced-order stochastic model is then compared to the spreading rate in the LES. The stochastic particle velocity representation is shown to reasonably reproduce spreading rates in sheared, precipitating, and non-precipitating cases using domain-averaged turbulent statistics from the LES. [ABSTRACT FROM AUTHOR]

We study a class of diffusion processes which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t). Under suitable conditions, we adopt Euler-Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we investigate convergence rates in the L2-norm the stochastic functional differential equation with infinite memory and random switching under the global Lipschitz conditions. Then we also discuss L2-convergence under the local Lipschitz conditions. [ABSTRACT FROM AUTHOR]

With the load center's continuous expansion and development of the AC power grid's scale and construction, the recipient grid under the multi-feed DC environment is facing severe challenges of DC commutation failure and bipolar blocking due to the high strength of AC-DC coupling and the low level of system inertia, which brings many complexities and uncertainties to economic scheduling. In addition, the large-scale grid integration of wind power, photovoltaic, and other intermittent energy sources makes the ultra-high-voltage (UHV) DC channel operation state randomized. The deterministic scenario-based timing power simulation is no longer suitable for the current complex and changeable grid operation state. In this paper, we first start with the description and analysis of the uncertainty in renewable energy (RE) sources, such as wind and solar, and reconstruct the time-sequence power model by using the stochastic differential equation model. Then, a carbon emission trading cost (CET) model is constructed based on the CET mechanism, and the two-stage locating and capacity optimization model for the UHV DC receiving end is proposed under the constraint of dispatch safety and stability. Among them, the first stage starts with the objective of maximizing the carrying capacity of the UHV DC receiving end grid; the second stage checks its dynamic safety under the basic and fault modes according to the results of the first stage and corrects the drop point and capacity of the UHV DC line with the objective of achieving safe and stable UHV DC operation at the lowest economic investment. In addition, the two-stage model innovatively proposes UHV DC relative inertia constraints, peak adjustment margin constraints, transient voltage support constraints under commutation failure conditions, and frequency support constraints under a DC blocking state. In addition, to address the problem that the probabilistic constraints of the scheduling model are difficult to solve, the discrete step-size transformation and convolution sequence operation methods are proposed to transform the chance-constrained planning into mixed-integer linear planning for solving. Finally, the proposed model is validated with a UHV DC channel in 2023, and the results confirm the feasibility and effectiveness of the model. [ABSTRACT FROM AUTHOR]

This paper explores a reinsurance and asset liability management problem in a factor model with dependent risks, considering economic factors as non-Markovian processes impacting the appreciation rate, volatility, and value of liabilities. It encompasses various stochastic volatility (SV) models like Hull-White, Heston, 3/2, and 4/2 models as special cases. The model addresses dependent risks through a common shock in claim number processes. The insurer's aim is to minimize variance in terminal net wealth while achieving a certain expected terminal net wealth, balancing reinsurance, new business, and asset allocation. The paper employs linear-quadratic (LQ) control and backward stochastic differential equation (BSDE) theory to derive both the efficient strategy and the efficient frontier. To illustrate our results, numerical examples are provided in two special cases, the 3/2 and 4/2 SV models. [ABSTRACT FROM AUTHOR]

Motivated by rational expectations theory, this paper studies a class of stochastic linear-quadratic dynamic optimisation problems involving a large number of weakly-coupled heterogeneous agents. Different from well-studied mean field games, these agents formalise a team with cooperation to minimise some social cost functional. Moreover, the state here evolves by some anticipated backward stochastic differential equation in which the terminal instead initial condition is specified and the anticipated terms are involved. Applying a so-called anticipated person-by-person optimality principle, we construct an auxiliary control problem for each agent based on decentralised information. The decentralised social strategy is derived by a class of new consistency condition systems, which are mean-field-type anticipated forward–backward stochastic differential delay equations (AFBSDDEs). The well-posedness of such consistency condition system is obtained via discounting method. The corresponding asymptotic social optimality is also verified as the population size goes to infinity. [ABSTRACT FROM AUTHOR]

This study introduces the fractional stochastic diffusion particle tracking model (FSD-PTM), a novel Lagrangian particle tracking model (PTM) designed to simulate stochastic suspended sediment transport by incorporating fractional Brownian motion (FBM). This FBM-based approach introduces memory into particle movements, particularly when particles are influenced by coherent turbulent structures near boundaries, such as ejection and sweep events. These coherent structures exhibit consistent motion over time, leading to correlated increments in particle movements. In scenarios unaffected by coherent structures, the FBM reverts to the Wiener process, preserving the independent increments. The FSD-PTM is applied to a steady, incompressible, fully developed two-dimensional open channel flow, and Monte Carlo simulations yield ensemble statistics results, including mean, variance, skewness of particle positions, particle velocity, and concentrations. These simulation results validate the FSD-PTM against experimental data, demonstrating its superior performance when considering correlated increments. Without such consideration, concentration profiles generated by previous PTMs fail to reach long-term equilibrium. A key innovation of the FSD-PTM is its integration of correlated motion directly within the governing stochastic differential equation (SDE), eliminating the need for additional terms. This model's flexibility also allows for the incorporation of various particle dynamic models, enhancing its utility for sediment transport analysis. The FSD-PTM represents a substantial advancement in modeling stochastic suspended sediment transport, particularly when particles' movements exhibit correlations due to surrounding turbulence. [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 study the convergence problem for mean field games with common noise and controlled volatility. We adopt the strategy recently put forth by Laurière and the second author, using the maximum principle to recast the convergence problem as a question of "forward-backward propagation of chaos" (i.e., (conditional) propagation of chaos for systems of particles evolving forward and backward in time). Our main results show that displacement monotonicity can be used to obtain this propagation of chaos, which leads to quantitative convergence results for open-loop Nash equilibria for a class of mean field games. Our results seem to be the first (quantitative or qualitative) that apply to games in which the common noise is controlled. The proofs are relatively simple and rely on a well-known technique for proving wellposedness of forward-backward stochastic differential equations, which is combined with displacement monotonicity in a novel way. To demonstrate the flexibility of the approach, we also use the same arguments to obtain convergence results for a class of infinite horizon discounted mean field games. Funding: J. Jackson is supported by the National Science Foundation [Grant DGE1610403]. L. Tangpi is partially supported by the National Science Foundation [Grants DMS-2005832 and DMS-2143861]. [ABSTRACT FROM AUTHOR]

Fuzzy fractional pantograph stochastic differential equations (FFPSDEs) is investigated here. The initial objective is to show the existence and uniqueness of solutions using Banach fixed point theorem. The second objective is discussing averaging principle of FFPSDEs, precisely, we will prove that the solutions of FFPSDEs can be approximated in the sense of mean square by the solutions of averaged fuzzy fractional stochastic system. [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]

AbstractThe Bass diffusion model is one of the most influential models for predicting the diffusion of new products and technologies, but the diffusion process will be affected by some uncertain factors. Usually, such kinds of uncertain factors are characterized as stochastic processes. However, in many cases, this approach can lead to uncertain issues. Therefore, in this article, a Bass diffusion model that incorporates the uncertainty theory is studied. First, we present an uncertain Bass diffusion model where the uncertain disturbance is modeled as an uncertain Liu process. Then, the existence and uniqueness of the uncertain Bass diffusion model are analyzed. Moreover, the uncertain Bass diffusion model is solved by using the 99-α-Runge-Kutta method. Finally, we carry out a numerical simulation on the output of the integrated circuit industry in China by using uncertain differential equations and stochastic differential equations. The results show that modeling with uncertain differential equations is superior to using stochastic differential equations. [ABSTRACT FROM AUTHOR]

AbstractWang, Chen, and Yang (2023) investigated the equilibrium behavioral strategy for a defined contribution (DC) pension plan during the accumulation phase and obtained the explicit expressions for the suboptimal equilibrium behavioral strategy. However, the authors failed to obtain the optimal solution and did not completely solve the pension investment problem with piecewise linear state-dependent risk tolerance. The primary objective of this article is to solve an improved version of the above DC pension plan problem thoroughly. We investigate the open-loop equilibrium portfolio for a DC pension plan and derive necessary and sufficient conditions for the open-loop equilibrium portfolio through coupled forward-backward stochastic differential equations (FBSDEs). The uniqueness of an equilibrium portfolio is obtained by decoupling FBSDEs. Finally, the economic significance and implications of the obtained open-loop equilibrium portfolio are illustrated from the theoretical perspective. [ABSTRACT FROM AUTHOR]

This paper studies the problem of exponential stability of second moments of large-scale distribution-dependent stochastic differential equations (SDEs, in short). By introducing isolated subsystems, under certain hypotheses added on the interconnected terms and using the moment estimation, exponential stability of second moments of isolated subsystems implies exponential stability of second moments of large-scale distribution-dependent SDEs. Specially, we provide sufficient and necessary conditions for the exponential stability of large-scale distribution-dependent SDEs with triangular structure and apply this results to study stabilization of large-scale distribution-dependent SDEs. Finally, the main results are illustrated by several examples and applications. [ABSTRACT FROM AUTHOR]

We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score-based MCMC method that is deterministic, resulting in a deterministic evolution for particles rather than a stochastic differential equation evolution. The score term is given in closed form by a regularized Wasserstein proximal, using a kernel convolution that is approximated by sampling. We demonstrate fast convergence on various problems and suggest improved dimensional dependence of mixing time bounds for the case of Gaussian distributions compared to the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally derive closed-form expressions for the distributions at each iterate for quadratic potential functions, characterizing the variance reduction. Empirical results demonstrate that the particles behave in an organized manner, lying on level set contours of the potential. Moreover, the posterior mean estimator of the proposed method is shown to be closer to the maximum a-posteriori estimator compared to ULA and MALA in the context of Bayesian logistic regression. Additional examples demonstrate competitive performance for Bayesian neural network training. [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]

This work focuses on a class of stochastic functional differential equations and neutral stochastic differential functional equations. By using a new approach, some sufficient conditions are obtained to guarantee the generalized mean square exponential stability for the equation under consideration. Certain existing results are refined and extended. Lastly, the validity of the main results is confirmed through several simulation examples. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method's capability to produce accurate and efficient solutions. [ABSTRACT FROM AUTHOR]

Economic growth modeling is one of the methods a government can use to formulate appropriate economic policies to improve the prosperity of its people. Differential equations and stochastic models play a major role in studying economic growth. This article aims to conduct a literature review on the use of differential equations in relation to stochastics to model economic growth. In addition, this article also discusses the use of differential and stochastic equations in economic growth models in Indonesia. This study involves searching for and selecting articles to obtain a collection of research works relevant to the application of differential and stochastic equations to economic growth models, supported by bibliometric analysis. The results of this literature review show that there is still little research discussing economic growth models using differential equations combined with stochastic models, especially those applied in Indonesia. While the application of these models remains relatively limited, their potential to offer deeper insights into the complex dynamics of economic growth is undeniable. By further developing and refining these models, we can gain a more comprehensive understanding of the factors driving growth and the potential implications of various economic policies. This will ultimately equip policy-makers with a more powerful analytical tool for making informed decisions. [ABSTRACT FROM AUTHOR]

Because noise is an inevitable attribute in gene regulatory networks, a stochastic differential equations model is constructed to study the impact of external noise on p53-Mdm2 regulatory network. Near the Hopf bifurcation of the corresponding deterministic model, external noise can induce stochastic resonance. When three stable steady states coexist in the corresponding deterministic model, external noise can lead to stochastic transition. Therefore, external noise can not only expand the parameter range of p53 regulatory network oscillations, but also increase the amplitude of p53 regulatory network oscillations, and is closely related to the outcome of cell fate. These findings deepen our understanding of the impact of external noise on gene regulatory networks and may provide new perspectives for the treatment of related diseases. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness of fully coupled forward-backward stochastic differential equations (FBSDEs in short) driven by anomalous sub-diffusions B L t under suitable monotonicity conditions on the coefficients. Here B is a Brownian motion on R and L t : = inf ⁡ { r > 0 : S r > t } , t ≥ 0 , is the inverse of a subordinator S with drift κ > 0 that is independent of B. Various a priori estimates on the solutions of the FBSDEs are also presented. [ABSTRACT FROM AUTHOR]

We focus on the dynamics and Wong-Zakai approximation for a class of retarded partial differential equations subjected to multiplicative white noise. We show that when restricted to a local region and under certain conditions, there exists a unique global solution for the truncated system driven by either the white noise or the approximation noise. Such solution generates a random dynamical system, and the solutions of Wong-Zakai approximations are convergent to solutions of the stochastic retarded differential equation. We also show that there exist invariant manifolds for the truncated system driven by either the white noise or the approximation noise, which are then the local manifolds for the untruncated systems, and prove that such invariant manifolds of the Wong-Zakai approximations converge to those of the stochastic retarded differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise. [ABSTRACT FROM AUTHOR]

We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems. [ABSTRACT FROM AUTHOR]

This paper presents stochastic stability and stochastic boundedness for certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is transformed into a neutral stochastic system of first-order equations and combined with a second-order quadratic function to derive a Lyapunov-Krasovskiĭ functional. This functional is then utilized to establish criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, new results on stochastic boundedness and uniform stochastic boundedness of solutions are derived. The results presented in this paper are original and improve upon existing literature. Two special cases of the theoretical results are provided to illustrate the practical application of the findings. [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise is investigated with consideration of a new method for dealing with singular integrals. Firstly, the estimate on higher moments for the solution is given. Secondly, under some suitable assumptions, we prove the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise by using the Hölder inequality. Finally, a simulation example is given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Uncertain differential equations, as an alternative to stochastic differential equations, have proved to be extremely powerful across various fields, especially in finance theory. The issue of parameter estimation for uncertain differential equations is the key step in mathematical modeling and simulation, which is very difficult, especially when the corresponding terms are driven by some complicated uncertain processes. In this paper, we propose the uncertainty counterpart of the threshold Ornstein–Uhlenbeck process in probability, named the uncertain threshold Ornstein–Uhlenbeck process, filling the gaps of the corresponding research in uncertainty theory. We then explore the parameter estimation problem under different scenarios, including cases where certain parameters are known in advance while others remain unknown. Numerical examples are provided to illustrate our method proposed. We also apply the method to study the term structure of the U.S. Treasury rates over a specific period, which can be modeled by the uncertain threshold Ornstein–Uhlenbeck process mentioned in this paper. The paper concludes with brief remarks and possible future directions. [ABSTRACT FROM AUTHOR]

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model's stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV. [ABSTRACT FROM AUTHOR]

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed. [ABSTRACT FROM AUTHOR]

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models. [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]

This paper explores the optimal management of defined contribution pension plans under multiple risks and strategic interactions using a stochastic differential game between two managers, accounting for model misspecification. Managers who are both competitive and averse to ambiguity can invest in a financial market that contains a risk-free asset, an inflation-linked bond, a market index, and a stock. These managers may have varying levels of aversion to ambiguous risks. The inflation index is computed using the Fisher equation. The market index and stock prices are described by a class of non-Markovian multivariate stochastic covariance models, with the market risk prices being reliant on a multivariate affine-diffusion factor process. Each manager aims to maximize the expected utility of inflation-adjusted terminal wealth relative to their competitors under the worst-case scenario of different measures, ensuring that the investment strategy is robust to model uncertainty. The robust Nash equilibrium investment strategies' explicit expressions, density generator processes under the well-defined worst-case scenarios, and corresponding value functions are derived using a backward stochastic differential equation approach to address this robust non-Markovian game. The admissibility of the robust equilibrium policies is confirmed under certain technical conditions. Finally, we provide numerical examples to demonstrate the effects of model parameters on robust investment policies and clarify the economic significance of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we study existence and uniqueness of solution stochastic differential equations involving fractional integrals driven by Riemann-Liouville multifractional Brownian motion and a standard Brownian. Then, we obtain approximate numerical solution of our problem and colon cancer chemotherapy effect model are presented to confirm our results. We show that considering time dependent Hurst parameters play an important role to get more realistic results. [ABSTRACT FROM AUTHOR]

We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index α ∈ [ 1 , 2) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of L p ([ 0 , T ] ; C b α + β (R d)) ∩ W 1 , p ([ 0 , T ] ; C b β (R d)) solution under the assumptions that the drift coefficient and nonhomogeneous term are in L p ([ 0 , T ] ; C b β (R d)) with p ∈ [ α / (α - 1) , + ∞ ] and β ∈ (0 , 1) . As applications, we prove the unique strong solvability as well as Davie's type uniqueness of time inhomogeneous stochastic differential equation with the drift in L p ([ 0 , T ] ; C b β (R d ; R d)) and driven by the α -stable process for β > 1 - α / 2 and p > 2 α / (α + 2 β - 2) . [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [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]

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Network inference has been extensively studied in several fields, such as systems biology and social sciences. Learning network topology and internal dynamics is essential to understand mechanisms of complex systems. In particular, sparse topologies and stable dynamics are fundamental features of many real-world continuous-time (CT) networks. Given that usually only a partial set of nodes are able to observe, we consider linear CT systems to depict networks since they can model unmeasured nodes via transfer functions. Additionally, measurements tend to be noisy and with low and varying sampling frequencies. This paper applies dynamical structure functions (DSFs) derived from linear stochastic differential equations (SDEs) to describe networks of measured nodes. A numerical sampling method, preconditioned Crank–Nicolson (pCN), is used to refine coarse-grained trajectories to improve inference accuracy. The proposed method can handle sparsely sampled data and unmeasurable nodes. Monte Carlo simulations indicate that the proposed method outperforms state-of-the-art methods with various network topologies. The developed method can be applied under a wide range of contexts, such as gene regulatory networks, social networks and communication systems. [ABSTRACT FROM AUTHOR]

Macrophages are essential components of the immune system's response to tumors, engaging in intricate interactions shaped by factors such as tumor type, progression, and the surrounding microenvironment. These dynamic relationships between macrophages and cancer cells have become a focal point of research, as scientists seek innovative ways to harness the immune system, including macrophages, for cancer immunotherapy. In this study, we introduce a novel model that examines the interaction between tumor and macrophage cells. We provide an in-depth analysis of the equilibrium points and their stability, as well as a thorough investigation into the solution properties of the model. Moreover, by incorporating a stochastic approach, we account for inherent randomness and fluctuations within the system, offering a more comprehensive understanding of tumor-immune dynamics. Numerical simulations further validate the model, providing key insights into how stochastic elements may influence tumor progression and immune response. [ABSTRACT FROM AUTHOR]

Guyon and Lekeufack (Quant. Finance 23:1221–1258, 2023) recently proposed a path-dependent volatility model and documented its excellent performance in fitting market data and capturing stylised facts. The instantaneous volatility is modelled as a linear combination of two processes; one is an integral of weighted past price returns and the other is the square root of an integral of weighted past squared volatility. Each weighting is built using two exponential kernels reflecting long and short memory. Mathematically, the model is a coupled system of four stochastic differential equations. Our main result is the wellposedness of this system: the model has a unique strong (non-explosive) solution for all parameter values. We also study the positivity of the resulting volatility process and the martingale property of the associated exponential price process. [ABSTRACT FROM AUTHOR]

Pseudo Marginal Metropolis-Hastings (PMMH) is a general approach to Bayesian inference when the likelihood is intractable, but can be estimated unbiasedly. Our article develops an efficient PMMH method that scales up better to higher dimensional state vectors than previous approaches. The improvement is achieved by the following innovations. First, a novel block version of PMMH that works with multiple particle filters is proposed. Second, the trimmed mean of the unbiased likelihood estimates of the multiple particle filters is used. Third, the article develops an efficient auxiliary disturbance particle filter, which is necessary when the bootstrap disturbance filter is inefficient, but the state transition density cannot be expressed in closed form. Fourth, a novel sorting algorithm, which is as effective as previous approaches but significantly faster than them, is developed to preserve the correlation between the logs of the likelihood estimates at the current and proposed parameter values. The performance of the sampler is investigated empirically by applying it to nonlinear Dynamic Stochastic General Equilibrium models with relatively high state dimensions and with intractable state transition densities and to multivariate GARCH diffusion-driven volatility in the mean models. Although we only apply the method to state space models, the approach will be useful in a wide range of applications such as large panel data models and stochastic differential equation models with mixed effects. [ABSTRACT FROM AUTHOR]

In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work. [ABSTRACT FROM AUTHOR]