Nonlinear Stochastic Differential Equations Research Articles

Stabilization by discrete‐time feedback control for highly nonlinear hybrid neutral stochastic functional differential equations with infinite delay

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example.

Adaptively tracking hypersonic gliding vehicles

The motion model plays a key role in tracking hypersonic gliding vehicles (HGVs). The process noise covariance matrix is used to scale the errors of the motion model, and its design has significant influence on the tracking accuracy. In this paper, a dynamics model with an adaptive process noise covariance matrix is proposed for tracking HGVs. The dynamics model is described by a nonlinear stochastic differential equation (SDE), and its corresponding process noise covariance matrix is calculated by adopting the modified Gauss-Legendre scheme. Furthermore, an adaptive adjustment method for process noise covariances of the augmented states is designed by analyzing the variation characteristics of the model errors. Specifically, the process noise covariances are adjusted by an adaptive factor: the square of the ratio of the estimated dynamic pressure to a prior dynamic pressure. Simulation results show that the proposed model can provide higher tracking accuracy than the dynamics model with fixed process noise covariances.

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations

Precision in disease dynamics: Finite difference solutions for stochastic epidemics with treatment cure and partial immunity

The complex and ever-changing characteristics of epidemic modelling, particularly when considering random elements, provide a substantial obstacle in creating precise and practical numerical methods for solving differential equations. This study contributes to this effort by introducing an innovative finite difference method for linear and non-linear stochastic and deterministic differential equations. This scheme expands explicitly upon the Euler Maruyama method, improving its precision for the deterministic aspect while ensuring coherence in dealing with stochastic terms. This contribution provides a numerical scheme that can be used to find solutions to linear and non-linear stochastic and deterministic differential equations. The scheme can be considered as the extension of the Euler Maruyama scheme for solving stochastic differential equations. The Euler Maruyama scheme offers a first-order accuracy of the deterministic model. Still, this scheme provides second-order accuracy for the deterministic part, whereas the integration of stochastic terms is the same in both schemes. The scheme is employed in a stochastic diffusive epidemic model with the effect of treatment, cure, and partial immunity. The comparison of the proposed scheme with the existing nonstandard finite difference method is made, and it is shown that the proposed scheme performs better than the nonstandard finite difference method in accuracy for the deterministic differential equations. It is also demonstrated that susceptible people rise whereas infected and recovered people decline by enhancing treatment cure rate. How does the cure rate of the treatment influence the number of the three populations, i.e., S(t), I(t), and R(t)? The results from the numerical simulation have provided useful insights into the dynamics of the epidemic model under various settings. This is particularly useful for influencing any public health plan and intervention. Thus, this work contributes numerical approaches and is an essential tool for epidemiological studies.

Trajectory fitting estimation for nonlinear stochastic differential equations with reflection

Trajectory fitting estimation for nonlinear stochastic differential equations with reflection

The Delayed Effect of Multiplicative Noise on the Blow-Up for a Class of Fractional Stochastic Differential Equations

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity.

Nonlinear stochastic model for epidemic disease prediction by optimal filtering perspective

Understanding and predicting novel diseases have become very important owing to the huge global health burden. The organization and study of mathematical models are critical in predicting disease behavior of the disease. In this paper, a new stochastic Susceptible‐Infected‐Recovered‐Death (SIRD) model for spreading epidemic disease is investigated. First, the deterministic SIRD model is considered, and then, by allowing randomness in the recovery and death rates that are not deterministic, the system of nonlinear stochastic differential equations is derived. For the suggested model, the existence and uniqueness of a positive global solution are demonstrated. The parameter estimation is done with the conditional least square estimator for deterministic models and the maximum likelihood estimator for stochastic ones. After that, we investigate a nonadditive state‐space model for spreading epidemic disease by considering infected as the hidden process variable. The problem of the hidden process variable from noisy observations is filtered, predicted, and smoothed using a recursive Bayesian technique. For estimating the hidden number of infected variables, closed‐form solutions are obtained. Finally, numerical simulations with both simulated and real data are performed to demonstrate the efficiency and accuracy of the current work.

Delay Feedback Control Based on Discrete-time State Observations for Highly Nonlinear Hybrid Stochastic Functional Differential Equations With Infinite Delay

Delay Feedback Control Based on Discrete-time State Observations for Highly Nonlinear Hybrid Stochastic Functional Differential Equations With Infinite Delay

Stochastic controls of fractional Brownian motion

Abstract We consider a stochastic control problem for a non-linear forward-backward stochastic differential equation driven by fractional Brownian motion, with Hurst parameter H ∈ ( 0 , 1 ) {H\in(0,1)} , in the case where the set of the control domain is convex. We provide an estimation of the solution and establish the necessary and sufficient optimality conditions in the form of the stochastic maximum principle. We apply the theory to solve a linear quadratic stochastic control problem.

Polarization dependent gain in Raman fiber amplifiers with multiple pumps.

In this paper, the polarization dependent gain (PDG) in Raman fiber amplifiers (RFAs) with multiple pumps is studied thoroughly. A comprehensive model, which takes the random polarization mode dispersion, the nonlinear coupling between the pumps, and the degree of polarization (DOP) of the pumps into account, is proposed. The complex nonlinear and random coupling inside the optical fiber is described by a set of nonlinear stochastic differential equations (SDEs), which enable co-simulation of the polarized part and the depolarized part of the multiple pumps. It is revealed that the average PDG and the PDG standard deviation are linearly proportional to the DOP of the pumps, which agrees with the single mode case. More importantly, when the pump wavelength is far away from the signal amplification range (pump-signal wavelength difference larger than 100 nm), its DOP still affects the PDG of the signal. Such a phenomenon is caused by the fact that the pumps interact with each other and the pump DOP could transfer among the pumps, which could enhance the PDG of the RFA. The findings in the work will have important implications for the design of the optical transmission systems with the multi-pump RFAs.

Global existence of dissipative solutions to the Camassa–Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

Simulating a strongly nonlinear backward stochastic partial differential equation via efficient approximation and machine learning

<abstract><p>We have studied a strongly nonlinear backward stochastic partial differential equation (B-SPDE) through an approximation method and with machine learning (ML)-based Monte Carlo simulation. This equation is well-known and was previously derived from studies in finance. However, how to analyze and solve this equation has remained a problem for quite a long time. The main difficulty is due to the singularity of the B-SPDE since it is a strongly nonlinear one. Therefore, by introducing new truncation operators and integrating the machine learning technique into the platform of a convolutional neural network (CNN), we have developed an effective approximation method with a Monte Carlo simulation algorithm to tackle the well-known open problem. In doing so, the existence and uniqueness of a 2-tuple adapted strong solution to an approximation B-SPDE were proved. Meanwhile, the convergence of a newly designed simulation algorithm was established. Simulation examples and an application in finance were also provided.</p></abstract>

A generalized finite element θ-scheme for backward stochastic partial differential equations and its error estimates

In this paper, we study numerical methods for solving a class of nonlinear backward stochastic partial differential equations. By utilizing finite element methods in space and θ-scheme in time, the proposed scheme forms a generalized spatio-temporal full discrete scheme, which can be solved in parallel. We rigorously prove the boundedness and error estimates, and obtain the optimal convergence rates in both time (first order/second order) and space (k + 1, k in L2 and H1, respectively). Numerical results are finally provided to demonstrate the effectiveness of the proposed scheme and validate the theoretical analyses.

Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations

Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations

A study of strong convergence of differential equations based on Euler’s algorithm

Abstract Differential equations have important applications in many fields, such as chemistry, biology, epidemiology, and finance. Most analytic solutions of differential equations are difficult to obtain. Therefore numerical solutions of differential equations become an important tool. The truncated Euler method is proposed in this paper, and we investigate how the truncated EM solution of the derived SDDE converges strongly under the local Lipschitz condition and the one-sided linear growth condition after relaxation. The basic strong convergence theorem is set up and the new notation X(t,x;s) is introduced as an analytic solution of the stochastic differential equation. Establish the assumption that the coefficients of the drift term and the coefficients of the diffusion term of the stochastic differential equation satisfy a contractionary monotonicity condition in order to prove, by induction, that the exact solution of this stochastic differential equation is bounded for a long time. Different examples are given to compare the simulated deviations between the numerical and analytical solutions of Euler’s algorithm and the modified Euler’s algorithm for the initial value problem of fractional order differential equations and to analyze the convergence of the two numerical solutions. The truncated Euler method is applied to highly nonlinear time-transformed stochastic differential equations by means of the dyadic principle, and it is proved that the order of convergence of the strong convergence of the truncated Euler-Maruyama method for time-transformed stochastic differential equations is min ( α , γ , 1 2 − ɛ ) \min \left( {\alpha ,\gamma ,{1 \over 2} - \varepsilon } \right) .

Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

Long-time behavior of stochastic Hamilton-Jacobi equations

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.

The exponential stability for locally monotone stochastic partial differential equations

This paper mainly studies the exponential stability for a class of nonlinear stochastic partial differential equations with locally monotone coefficients. The generalized variational framework is adopted to analyze the mean square exponential stability and pathwise exponential stability. Additionally, the use of multiplicative Itô noise with sufficient strength to enhance the stability of solution to stochastic partial differential equation is explored when the mean square exponential stability does not hold.

Sticky nonlinear SDEs and convergence of McKean–Vlasov equations without confinement

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

Delay feedback control of highly nonlinear neutral stochastic delay differential equations driven by G-Brownian motion

Delay feedback control of highly nonlinear neutral stochastic delay differential equations driven by G-Brownian motion