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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]

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]

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 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]

We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type. [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]

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]

We use the lens of weak signal asymptotics to study a class of sequentially randomized experiments, including those that arise in solving multiarmed bandit problems. In an experiment with n time steps, we let the mean reward gaps between actions scale to the order 1/n to preserve the difficulty of the learning task as n grows. In this regime, we show that the sample paths of a class of sequentially randomized experiments—adapted to this scaling regime and with arm selection probabilities that vary continuously with state—converge weakly to a diffusion limit, given as the solution to a stochastic differential equation. The diffusion limit enables us to derive refined, instance-specific characterization of stochastic dynamics and to obtain several insights on the regret and belief evolution of a number of sequential experiments including Thompson sampling (but not upper-confidence bound, which does not satisfy our continuity assumption). We show that all sequential experiments whose randomization probabilities have a Lipschitz-continuous dependence on the observed data suffer from suboptimal regret performance when the reward gaps are relatively large. Conversely, we find that a version of Thompson sampling with an asymptotically uninformative prior variance achieves near-optimal instance-specific regret scaling, including with large reward gaps, but these good regret properties come at the cost of highly unstable posterior beliefs. This paper was accepted by Baris Ata, stochastic models and simulation. Supplemental Material: The data and online appendix are available at https://doi.org/10.1287/mnsc.2023.4964. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate general mean-field backward stochastic differential equations (MFBSDEs) in multi-dimension with diagonally quadratic generators $ f(\omega,t,y,z,\mu) $, that is, the coefficients depend not only on the solution processes $ (Y,Z) $, but also on their law $ \mathbb{P}_{(Y,Z)} $, as well as they have a diagonally quadratic growth in $ Z $ and super-linear growth (or even a quadratic growth) in the law of $ Z $ and this latter fact is totally novel. We start by establishing through a fixed point theorem the existence and the uniqueness of local solutions in the 'Markovian case' $ f(t,Y_{t},Z_{t},\mathbb{P}_{(Y_{t},Z_{t})}) $ when the terminal value is bounded. Afterwards, global solutions are constructed by stitching local solutions. Finally, employing the $ \theta $-method, we explore the existence and the uniqueness of global solutions for diagonally quadratic mean-field BSDEs with convex generators, even in the case of unbounded terminal values that have exponential moments of all orders. These results are extended to a Volterra-type case where the coefficients can even be of quadratic growth with respect to the law of $ Z $, not only with diagonally quadratic growth in $ Z $ like in existing literatures, and this is completely new even for non-mean-field case. [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]

The stability and Hopf bifurcation of a class of flywheel governor systems subject to stochastic excitation under time-delay feedback are studied. The model is simplified using center manifold reduction to obtain a paradigm for a two-dimensional central subsystem. The system converges asymptotically to an Itô stochastic differential equation based on polar transformations and stochastic averaging method. The stochastic stability of the system is analyzed by applying the maximum Lyapunov exponent and the singular boundary theory, and its probability density function is calculated. In addition, it is found through Monte Carlo numerical simulations that the effects of noise intensity and time-delay on the stochastic P-bifurcation of the system are opposite, and proper noise intensity makes the flywheel governor system converge to the equilibrium position, while a large time-delay leads to stochastic bifurcation and makes the flywheel governor system unstable. Finally, the two-parameter bifurcation diagram is used to further analyze the impact of time-delay and noise intensity on the bifurcation of the system under variable load torque. It is found that the influence of noise intensity and time-delay on the periodic oscillation state is the same, and the stability of the flywheel governor system will decrease with both of their increases. This also further demonstrates the duality of the impact of noise intensity on the same system under different conditions. The above research provides better theoretical references for the design and safe operation of speed control systems. [ABSTRACT FROM AUTHOR]

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion. [ABSTRACT FROM AUTHOR]

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modeled by a stochastic differential equation of Stratonovich-type driven by semimartingales on the orthonormal frame bundle. The stability of the system is examined via large deviations. We prove the large deviation principles for the projection curves on the base manifold and their horizontal lifts, respectively, provided that the large deviation holds for the random Euclidean curves as semimartingales. The large deviation results for the case of compact manifolds and two special cases of non-compact manifolds are established. [ABSTRACT FROM AUTHOR]

AbstractWe 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]

The mode coupling in a graded-index polymer photonic crystal fiber (GI PPCF) with a solid core has been investigated using the Langevin equation. Based on the computer-simulated Langevin force, the Langevin equation is numerically integrated. The numerical solutions of the Langevin equation align with those of the time-independent power flow equation (TI PFE). We showed that by solving the Langevin equation, which is a stochastic differential equation, one can successfully treat a mode coupling in GI PPCFs, which is an intrinsically stochastic process. We demonstrated that, in terms of effectiveness, the Langevin equation is preferable compared to the TI PFE. The GI PPCF achieves the equilibrium mode distribution (EMD) at a coupling length that is even shorter than the conventional GI plastic optical fiber (POF). The application of multimode GI PCFs in communications and optical fiber sensor systems will benefit from these findings. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model. [ABSTRACT FROM AUTHOR]

This article is devoted towards some necessary conditions for the existence of the mild solution of 'second‐order semilinear stochastic delay differential equations' having a variable delay in the state with the help of Banach fixed point theorem and theory of cosine family which is strongly continuous. In the last section, we have discussed an example to understand the theory of provided results in a better manner. [ABSTRACT FROM AUTHOR]

Stochastic delayed modeling has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. The covid-19 has been controlled globally but there is still a threat and appears in different variants like omicron and SARS-CoV-2 etc. globally. This article, considered pattern a mathematical model based on Susceptible, Infected, and recovered populations with highly nonlinear incidence rates. we studied the dynamics of the coronavirus model; a newly proposed version is a stochastic delayed model that is based on nonlinear stochastic delayed differential equations (SDDEs). Transition probabilities and parametric perturbation methods were used for the construction of the stochastic delayed model. The fundamental properties like positivity, boundedness, existence and uniqueness, and stability results of equilibria of the model with certain conditions of reproduction number are studied regularly. Also, the extinction and persistence of disease are studied with the help of well-known theorems. The numerical methods used to find a visualization of results due to the complexity of stochastic delayed differential equations. Furthermore, for computational analysis, we implemented existing methods in the literature and compared their results with the proposed method like nonstandard finite difference for stochastic delayed model. The proposed method restores all dynamical properties of the model with a free choice of time steps. [ABSTRACT FROM AUTHOR]

Angular random walk (ARW), rate random walk (RRW), and bias instability (BI) are the main noise types in inertial measurement units (IMUs) and thus determine the navigation performance of IMUs. BI is the flicker noise, which determines the noise level of an inertial sensor. The traditional error modeling approach involves modeling the ARW and BI processes as RRW or Gauss–Markov (GM) processes, and this approach is applied as a suboptimal filter in the global navigation satellite system (GNSS)/inertial navigation system (INS) extended Kalman filter (EKF). In this paper, the random error identification processes for white noise and colored noise for inertial sensors are separated using the Allan variance and power spectral density methods and the equivalence of the stochastic process differential equations of bias instability and a combination of multiple first-order GM processes are derived. A colored noise compensation method is proposed based on the enhanced EKF model. Experimental results demonstrate that, compared to traditional error models, our proposed model reduces positional drift error in dynamic testing from 195 to 49 m, enhancing positional accuracy by 40.2%. These findings confirm the potential and superiority of our method in complex navigation environments. [ABSTRACT FROM AUTHOR]

Motivation: Learning cellular dynamics through reconstruction of the underlying cellular potential energy landscape (aka Waddington landscape) from time-series single-cell RNA sequencing (scRNA-seq) data is a current challenge. Prevailing data-driven computational methods can be hampered by the lack of physical principles to guide learning from complex data, resulting in reduced prediction accuracy and interpretability when applied to infer cell population dynamics. Results: Here, we propose PI-SDE, a physics-informed neural stochastic differential equation (SDE) framework that combines the Hamilton–Jacobi (HJ) equation and neural SDE to learn cellular dynamics. Grounded in potential energy theory of biological systems, PI-SDE integrates the principle of least action by enforcing the HJ equation when reconstructing cellular potential energy function. This approach not only facilitates accurate predictions, but also improves interpretability, especially in the reconstructed potential energy landscape. Through benchmarking on two real scRNA-seq datasets, we demonstrate the importance of incorporating the HJ regularization term in dynamic inference, especially in predicting gene expression at held-out time points. Meanwhile, the learned potential energy landscape provides biologically interpretable insights into the process of cell differentiation. Our framework enhances model performance, while maintaining robustness and stability. Availability: PI-SDE software is available at https://github.com/QiJiang-QJ/PI-SDE. [ABSTRACT FROM AUTHOR]

Background: The regulatory mechanisms guiding CD4 T cell differentiation are complex and are further influenced by intrinsic cell variability along with that of microenvironmental cues, such as cytokine and nutrient availability. Objective: This study aims to expand our understanding of CD4 T cell differentiation by examining the influence of intrinsic noise on cell fate. Methodology: A model based on a complex regulatory network of early signaling events involved in CD4 T cell activation and differentiation was described in terms of a set of stochastic differential equation to assess the effect of noise intensity on differentiation efficiency to the Th1, Th2, Th17, Treg, and TFH effector phenotypes under defined cytokine and nutrient conditions. Results: The increase of noise intensity decreases differentiation efficiencies. In a microenvironment of Th1-inducing cytokines and optimal nutrient conditions, noise levels of 3%, 5% and 10% render Th1 differentiation efficiencies of 0.87, 0.76 and 0.62, respectively, underscoring the sensitivity of the network to random variations. Further increments of noise reveal that the network is relatively stable until noise levels of 20%, where the resulting cell phenotypes becomes heterogeneous. Notably, Treg differentiation showed the highest robustness to noise perturbations. A combined Th1-Th2 cytokine environment with optimal nutrient levels induces a dominant Th1 phenotype; however, removal of glutamine shifts the balance towards the Th2 phenotype at all noise levels, with an efficiency similar to that obtained under Th2-only cytokine conditions. Similarly, combinations of Th1/Treg and Treg/Th17-inducing cytokines along with the removal of either tryptophan or oxygen shift the dominant Th1 and Treg phenotypes towards Treg and Th17 respectively. Model results are consistent with differentiation efficiency patterns obtained under wellcontrolled experimental settings reported in the literature. Conclusion: The stochastic CD4 T cell mathematical model presented here demonstrates a noise-dependent modulation of T cell differentiation induced by cytokines and nutrient availability. Modeling results can be explained by the network topology, which assures that the system will arrive at stable states of cell functionality despite variable levels of biological intrinsic noise. Moreover, the model provides insights into the robustness of the T cell differentiation process. [ABSTRACT FROM AUTHOR]

This paper investigates an optimal investment‐reinsurance problem for an insurer who possesses inside information regarding the future realizations of the claim process and risky asset process. The insurer sells insurance contracts, has access to proportional reinsurance business, and invests in a financial market consisting of three assets: one risk‐free asset, one bond, and one stock. Here, the nominal interest rate is characterized by the Vasicek model, and the stock price is driven by Heston's stochastic volatility model. Applying the enlargement of filtration techniques, we establish the optimal control problem in which an insurer maximizes the expected power utility of the terminal wealth. By using the dynamic programming principle, the problem can be changed to four‐dimensional Hamilton–Jacobi–Bellman equation. In addition, we adopt a deep neural network method by which the partial differential equation is converted to two backward stochastic differential equations and solved by a stochastic gradient descent‐type optimization procedure. Numerical results obtained using TensorFlow in Python and the economic behavior of the approximate optimal strategy and the approximate optimal utility of the insurer are analyzed. [ABSTRACT FROM AUTHOR]

This paper is motivated by the paper [2]. The main aim of this paper is to extend the stability result from [16], related to the θ-Euler- Maruyama method (θ ∈ (1/2, 1)) for a class of neutral stochastic differential equations with time-dependent delay. The theta method is defined such that, in general case, it is implicit in both drift coefficient and neutral term. Sufficient conditions of the a.s. exponential stability of the θ-Euler-Maruyama method, including the linear growth condition on the drift coe_cient of the equation, are revealed. The stability result is established for larger class of neutral terms than that considered in the second cited paper. An example is provided to support the main results of the paper. [ABSTRACT FROM AUTHOR]

This study deals with the synchronization problem of the stochastic fractional-order biomathematical model of muscular blood vessels (SFOMBVs) with nonlinear inputs using a state-feedback adaptive control law, in which the effects of uncertainties and external disturbances are considered in the modeling. The linear state feedback and adaptive laws are combined via an adaptive update law, leading to a simple and robust adaptive feedback scheme. The synchronization and tracking performances of the SFOMBV master and slave systems are investigated, and the stability of the proposed control system is proved based on stochastic analysis techniques and Lyapunov theory. The simulation results show that the proposed control method can lead an abnormal muscular blood vessel to a normal trajectory with good robustness by reducing the tracking error. [ABSTRACT FROM AUTHOR]

It is known that time delays generally make a system unstable. However, it is numerically observed that the diffusion coefficients of the Wiener processes with time-delayed feedback decrease while increasing the time delay τ. In particular, the decay of the diffusion coefficients with the form ( 1 1 + τ ) 2 has been confirmed by numerical simulations [Ando et al., Phys. Rev. E 96, 012148 (2017)]. In this paper, we present two analytical derivations for the relation ( 1 1 + τ ) 2 by dynamical system approaches using the Laplace transform and stochastic differential equations. [ABSTRACT FROM AUTHOR]

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1 < q < 2 . Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be $ \alpha -\frac {1}{2} $ α − 1 2 with $ \frac {1}{2} \lt \alpha \lt 1 $ 1 2 < α < 1. Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method. [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]

This paper presents a novel approach to identify an optimal coefficient for evaluating a player's batting average, strike rate, and bowling average, aimed at achieving an optimal team score through dynamic modeling using a path integral method. Additionally, it introduces a new model for run dynamics, represented as a stochastic differential equation, which factors in the average weather conditions at the cricket ground, the specific weather conditions on the match day (including abrupt changes that may halt the game), total attendance, and home field advantage. An analysis of real data is been performed to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov's theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering. [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]

In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit. [ABSTRACT FROM AUTHOR]

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example. [ABSTRACT FROM AUTHOR]

This paper implements the stochastic filters for state and parameter estimation of nonuniform transmission lines (NTL). In general, transmission line (TL) problem is a continuous time and space problem. By taking the line loading noise into account, the TL equations become a stochastic partial differential equation (PDE) rather than a simple set of coupled finite stochastic differential equations (SDE). By transforming the spatial variables into the Fourier domain, the stochastic PDE can be transformed into an infinite sequence of SDE. After truncation to a finite set of Fourier series coefficients, it becomes a finite set of coupled linear SDE, which is the required domain in which extended Kalman filter (EKF) and unscented Kalman filter (UKF) can be applied. For state space equation of EKF and UKF, the voltage and current of periodic NTL are expanded into an infinite set of spatial harmonics. In this way, the voltage and current measurement of NTL become an eigenvalue problem. The NTL is considered as cascade of small infinite NTL and the four distributed primary parameters of the periodic NTL are expressed using Fourier series expansion. Finally, the Kalman filter (KF)-based state estimation and the EKF- and UKF-based parameter estimation have been compared with recursive least squares (RLS) method. The simulation results present the superiority of the method. [ABSTRACT FROM AUTHOR]

Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical solutions, accounting for the mutual influences between forward and backward processes, and considering the relationship with respect to SPDEs rather than PDEs. This paper introduces, for the first time, a numerical method for solving multi-dimensional coupled FBDSDEs. By integrating an optimal control-based approach with deep neural networks, it effectively addresses the coupling-related challenges between forward and backward equations. Computational examples of coupled FBDSDEs with explicit solutions demonstrate that the proposed deep learning-based numerical algorithm achieves commendable performance in terms of both accuracy and efficiency. [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]

In this article, we study the optimal control of stochastic differential equations with random impulses. We optimize the performance index and add the influence of random impulses to the performance index with a random compensation function. Using the idea of stochastic analysis and dynamic programming principle, a new Hamilton–Jacobi–Bellman (HJB) equation is obtained, and the existence and uniqueness of its viscosity solution are proved. [ABSTRACT FROM AUTHOR]

In this paper we study the drift parameter estimation for reflected stochastic linear differential equations of a large signal. We discuss the consistency and asymptotic distributions of trajectory fitting estimator (TFE). [ABSTRACT FROM AUTHOR]