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This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend. [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]

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 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]

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]

We present a neural network approach for approximating the value function of high-dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semilinear parabolic partial differential equations and forward-backward stochastic differential equations. To focus the sampling on relevant states during neural network training, we use the stochastic Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. By design, our approach coincides with the method of characteristics for the non-viscous Hamilton--Jacobi--Bellman equation arising in deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem. Our numerical experiments highlight our scheme's ability to identify the relevant parts of the state space and produce meaningful value estimates. Using a two-dimensional model problem, we demonstrate the importance of the stochastic PMP to inform the sampling and compare it to a finite element approach. With a nonlinear control affine quadcopter example, we illustrate that our approach can handle complicated dynamics. For a 100-dimensional benchmark problem, we demonstrate that our approach improves accuracy and time-to-solution, and, via a modification, we show the wider applicability of our scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with the discrete-time infinite horizon backward stochastic differential equation, i.e., the infinite horizon backward stochastic difference equation. We first prove the existence and uniqueness of the solution of this equation. Then the discrete-time stochastic recursive control problem is studied, in which the cost functional is set to be the solution of the infi- nite horizon backward stochastic difference equation. By introducing a proper discrete-time infinite horizon dual equation, we prove the stochastic maximum principle and the verification theorem for this recursive control problem. Finally we apply the derived stochastic maximum principle to the optimal consumption problem arising from a type of long-term trust fund. [ABSTRACT FROM AUTHOR]

This work addresses stochastic integro-delay differential equations (SIDDEs) of second order with two constant delays. In the study, two new results including sufficient conditions on stochastic asymptotic stability and stochastic boundedness in probability of solutions of the given SIDDEs are proved. The proofs of new results are done by using a Lyapunov-Krasovskii functional (L-KF) as a basic tool. To demonstrate the validity of the obtained results, two examples are provided. According to a comparison with previous literature, the results of this study are new and also allow new contributions to the qualitative theory of SIDDES. [ABSTRACT FROM AUTHOR]

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth. [ABSTRACT FROM AUTHOR]

In this article, we consider the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay and reveal conditions of the divergence of the p-th absolute moments of the corresponding approximate solutions when p ∈ (0 , ∞). This implies the strong Lp-divergence of the method at finite time, for p ∈ [ 1 , + ∞) , and shows that its numerically weak convergence fails to hold. This article is motivated by the paper Hutzenthaler, M., Jentzen, A., Kloeden, P. E.: Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A 467 (2011), no. 2130, 1563–1576, where a class of ordinary stochastic differential equations with superlinearly growing coefficients is studied. [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]

In this paper, we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays. The $ \epsilon $ -Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the consistency condition. By means of variational analysis, the system of consistency conditions can be expressed by forward-backward stochastic differential equations. Numerical examples illustrate the sensitivity of solutions of advanced backward stochastic differential equations to time delays, the effect of the the population's collective behaviors, and the consistency of mean-field estimates. [ABSTRACT FROM AUTHOR]

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). The goal of this work is twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver $ A $. Second, we study penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a constrained optimal stopping problem or a constrained Dynkin game in which the set of minimizer's admissible exercise times is constrained to the right support of the measure generated by $ A $. [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]

We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer who wishes to maximize the expected power utility of its terminal wealth on finite time horizon. We are also interested in the problem of maximizing the growth rate of expected power utility per unit time on the infinite time horizon. The risk process of the insurer is described by an approximation of the classical Cramér–Lundberg process. The insurer invests in a market consisting of a bank account and multiple risky assets. The mean returns of the risky assets depend linearly on economic factors that are formulated as the solutions of linear stochastic differential equations. With this setting, Hamilton–Jacobi–Bellman equations that are derived via a dynamic programming approach have explicit solution obtained by solving a matrix Riccati equation. Hence, the optimal investment and reinsurance strategies can be constructed explicitly. Finally, we present some numerical results related to properties of our optimal strategy and the ruin probability using the optimal strategy. [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 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]

Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob. 29 , 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul. 17 , 2019). In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one. [ABSTRACT FROM AUTHOR]

Fractional Stochastic Differential Equations are becoming more popular in the literature as they can model phenomena in financial data that typical Stochastic Differential Equations models cannot. In the formulation considered here, the Hurst parameter, H, controls the Fraction of Differentiation, which needs to be estimated from the data. Fortunately, the covariance structure among observations in time is easily expressed in terms of the Hurst parameter which means that a likelihood is easily defined. This work derives the Maximum Likelihood Estimator for H, which shows that it is biased and is not a consistent estimator. Simulation data used to understand the bias of the estimator is used to create an empirical bias correction function and a bias-corrected estimator is proposed and studied. Via simulation, the bias-corrected estimator is shown to be minimally biased and its simulation-based standard error is created, which is then used to create a 95% confidence interval for H. A simulation study shows that the 95% confidence intervals have decent coverage probabilities for large n. This method is then applied to the S&P500 and VIX data before and after the 2008 financial crisis. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDE) are a powerful tool to model biological regulatory processes with intrinsic and extrinsic noise. However, numerical simulations of SDE models may be problematic if the values of noise terms are negative and large, which is not realistic for biological systems since the molecular copy numbers or protein concentrations should be non-negative. To address this issue, we propose the composite Patankar-Euler methods to obtain positive simulations of SDE models. A SDE model is separated into three parts, namely, the positive-valued drift terms, negative-valued drift terms, and diffusion terms. We first propose the deterministic Patankar-Euler method to avoid negative solutions generated from the negative-valued drift terms. The stochastic Patankar-Euler method is designed to avoid negative solutions generated from both the negative-valued drift terms and diffusion terms. These Patankar-Euler methods have the strong convergence order of a half. The composite Patankar-Euler methods are the combinations of the explicit Euler method, deterministic Patankar-Euler method, and stochastic Patankar-Euler method. Three SDE system models are used to examine the effectiveness, accuracy, and convergence properties of the composite Patankar-Euler methods. Numerical results suggest that the composite Patankar-Euler methods are effective methods to ensure positive simulations when any appropriate stepsize is used. [ABSTRACT FROM AUTHOR]

In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of { X ε } 0 < ε ⩽ 1 in C ([ 0 , T ] ; R n). Subsequently, we demonstrate that any accumulation point of { X ε } 0 < ε ⩽ 1 can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider a class of linear quadratic extended mean field games (MFGs) with common noises where the state coefficients and the cost functional vary with the mean field term in a nonlinear way. Based on stochastic maximum principle, solving the mean field game is transformed into solving a conditional mean field forward-backward stochastic differential equation (FBSDE). We first establish solvability for a type of (more general) conditional mean field FBSDEs under monotonicity conditions. We further provide some regularity results which lead to classical solutions for the associated master equations. In particular, the linear quadratic extended mean field game is solved and classical solution for (extended mean field game) master equation is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov's and Schaefer's fixed point theorems. An example illustrating our result is provided. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results. [ABSTRACT FROM AUTHOR]

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

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte–Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein–Uhlenbeck and a stochastic Prandtl–Tomlinson bath model are considered. [ABSTRACT FROM AUTHOR]

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n ∈ N and truncation dimension parameter M ∈ N , is of the order n − 1 / 2 + δ (M) such that δ (⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported. [ABSTRACT FROM AUTHOR]