Nonlinear Stochastic Differential Equations Research Articles

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter.

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

Solving nonlinear stochastic differential equations via fourth-degree hat functions

The article focuses on introducing novel fourth-degree hat functions (FDHFs) for constructing new operations matrices used to find numerical solutions for nonlinear stochastic differential equations (NSDEs). The technique’s effort is summarized by utilizing FDHFs to construct operations matrices that transform the given problem into nonlinear algebraic equations. The advantage of this technique lies in its simplicity for calculating the unknown coefficients of the function’s approximation without requiring any integration. As a result, the proposed approach incurs low computational expenses. Additionally, an error analysis of this approach was conducted, demonstrating a convergence rate of O(h5). Several examples were implemented to support and illustrate the effectiveness and capability of the proposed technique.

A microscopic model of black hole evaporation in two dimensions

We present a microscopic model of black hole (BH) ‘evaporation’ in asymptotically AdS2 spacetimes dual to the low energy sector of the SYK model. To describe evaporation, the SYK model is coupled to a bath comprising of Nf free scalar fields Φi. We consider a linear combination of couplings of the form OSY K(t)∑iΦi(0, t), where OSY K involves products of the Kourkoulou-Maldacena operator iJ/N{sum}_{k=1}^{N/2}{s}_k^{prime }{psi}_{2k-1}(t){psi}_{2k}(t) specified by a spin vector s′. We discuss the time evolution of a product of (i) a pure state of the SYK system, namely a BH microstate characterized by a spin vector s and an effective BH temperature TBH, and (ii) a Calabrese-Cardy state of the bath characterized by an effective temperature Tbath. We take Tbath ≪ TBH, and TBH much lower than the characteristic UV scale J of the SYK model, allowing a description in terms of the time reparameterization mode. Tracing over the bath degrees of freedom leads to a Feynman-Vernon type effective action for the SYK model, which we study in the low energy limit. The leading large N behaviour of the time reparameterization mode is found, as well as the Oleft(1/sqrt{N}right) fluctuations. The latter are characterized by a non-Markovian non-linear stochastic differential equation with non-local Gaussian noise. In a restricted range of couplings, we find two classes of solutions which asymptotically approach (a) a BH at a lower temperature, and (b) a horizonless geometry. We identify these with partial and complete BH evaporation, respectively. Importantly, the asymptotic solution in both cases involves the scalar product of the spin vectors s.s′, which carries some information about the initial state. By repeating the dynamical process O(N2) times with different choices of the spin vector s′, one can in principle reconstruct the initial BH microstate.

Stabilization of highly nonlinear hybrid neutral stochastic differential equations with multiple time-varying delays and different structures

In this paper, the stabilization of a class of hybrid differently structured neutral stochastic differential equations with multiple time-varying delays is taken into consideration. The coefficients of these hybrid neutral stochastic differential delay equations with Markov switching (hybrid NSDDEs) are highly nonlinear in some modes while satisfying linear growth criteria in others. To ensure the existence and uniqueness of the solution, mild conditions are first applied to the system coefficients. Second, design techniques for discrete-time feedback control for only some of the modes are offered in order to stabilize the above systems that are unstable. New M-matrix criteria that are simple to calculate and verify are also proposed for the control functions. Then, a new theory on asymptotic and exponential stabilization is developed by using the Lyapunov function technique. Finally, the results are demonstrated with an example.

A Martingale Approach to Asymptotic Stability Of Nonlinear Stochastic Difference Equations With Bounded Noise in R1

A Martingale Approach to Asymptotic Stability Of Nonlinear Stochastic Difference Equations With Bounded Noise in R1

Analysis of Brownian motion in stochastic Schrödinger wave equation using Sardar sub-equation method

In this paper, the stochastic nonlinear Schrödinger equation (NLSE) forced by multiplicative noise that represents the temporal change of fluctuations is examined in a more extended form. Symmetry reduction (nonclassical symmetry) is used to convert the nonlinear partial differential equation into a nonlinear ordinary differential equation to obtain new optical soliton solutions. The Sardar sub-equation method is suggested to solve this stochastic nonlinear problem. The obtained stochastic solitons illustrate the dispersion of waves in optical fiber transmissions. This approach enables us to study a wide range of solutions with significant physical perspectives, including dark solitons, bright solitons, singular and periodic solitons with their noise term effects involved in Brownian motion based on the Itô sense. The noise term effects on the solitons are presented using 3D and 2D graphs. The results and calculations show the significance, accuracy and efficiency of the method. Various stable and unstable nonlinear stochastic differential equations that are found in mathematics, physics and other applied areas can be solved using this technique.

Computational analysis of the coronavirus epidemic model involving nonlinear stochastic differential equations

Stochastic methods significantly solve stochastic differential equations such as stochastic equations with a delay, stochastic fractional and fractal equations, stochastic partial differential equations, and many more. The coronavirus is still a threat to humans and puts people in danger. The model is a symmetric and compatible distribution family. In this case, the present model contains seven sub-populations of humans: susceptible, exposed, infected, quarantined, vaccinated, recovered, and dead. Two deterministic to stochastic formation types are studied, namely, transition probabilities and nonparametric perturbations. The positivity and boundedness of the stochastic model are analyzed. The stochastic Euler, stochastic Runge–Kutta, and Euler–Maruyama methods solve the stochastic system. Unfortunately, many issues originate, such as negativity, boundedness, and violation of dynamical consistency. The nonstandard finite difference method is designed in the sense of stochasticity to restore the dynamic properties of the model. In the end, simulations are carried out in contrast to deterministic and stochastic solutions. Overall, our findings shed light on the underlying mechanisms of COVID-19 dynamics and the influence of environmental factors on the spread of the disease, which can help make informed policy decisions and public health interventions.

Boundedness and asymptotic Behaviour of Solutions of some second-order nonlinear stochastic differential equations with delay

This paper considers a certain second-order nonlinear stochastic differential equation with delay. Novel conditions for the existence of solutions that are uniformly bounded and ultimately bounded are obtained. Moreover, we also study the asymptotic behaviour of solutions for the considered equation. We employ Lyapunov’s second method via an appropriate complete Lyapunov functional to achieve these. Obtained results are new, and they improve and complement some existing relatively recent results in the literature. Finally, an example is provided to illustrate the obtained results.

Numerical solution of nonlinear stochastic differential equations with fractional Brownian motion using fractional-order Genocchi deep neural networks

In this work, a new computational scheme namely fractional-order Genocchi deep neural network (FGDNN) is introduced to solve a class of nonlinear stochastic differential equations (NSDEs) driven by fractional Brownian motion (FBM) with Hurst parameter H∈(0,1). For generating the fractional Brownian motion, derivative and fractional-order integral operational matrices based on the fractional-order Genocchi functions and the classic Brownian motion approximation with help of the Gauss–Legendre quadrature are obtained. The FGDNN method is utilized the fractional-order Genocchi functions and Tanh function as activation functions of the deep structure. By considering deep neural network’s ability in approximating a nonlinear function, we present a new approximate function to estimate unknown function. This approximate function contains the FGDNN with unknown weights. Using the classical optimization method and Newton’s iterative scheme, the weights are adjusted such that the approximate function satisfies the under study problem. The convergence analysis of the mentioned scheme is discussed. Finally, some illustrative examples are included to show the applicability, accuracy and efficiency of the new method. The FGDNN method is compared with the analytical solutions and the numerical results obtained through the Chebyshev cardinal wavelets and hat functions methods.

Computing non-equilibrium trajectories by a deep learning approach

Predicting the occurrence of rare and extreme events in complex systems is a fundamental problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in recent years, which better estimate tail distributions. They often use large deviation concepts without the need to perform heavy direct ensemble simulations. In particular, a well-known approach is to derive a minimum action principle and find its minimizers.The analysis of rare reactive events in non-equilibrium systems without detailed balance is notoriously difficult both theoretically and computationally. They can often be described in the limit of small noise by the Freidlin-Wentzell action. Here, we propose a method which minimizes the geometric action instead using neural networks: it is called deep gMAM. It relies on a natural and simple machine-learning formulation of the classical gMAM approach when the Lagrangian is known. We provide a detailed description of the method as well as many examples: from bimodal switches in nonlinear stochastic (partial) differential equations, including a codimension-1 nucleation in an Allen-Cahn-Ginzburg-Landau 5-D stochastic partial differential equation, to quasi-potential estimates and extreme events in Burgers turbulence.

Periodically intermittent discrete observations control for stabilization of highly nonlinear hybrid stochastic differential equations

In this paper, the stabilization of a class of hybrid stochastic systems whose drift coefficients satisfy high nonlinear growth conditions is studied. A periodically intermittent controller based on discrete time state observations is designed. The core idea of this paper is to derive the stability of the controlled system by constructing Lyapunov function. An example is given to prove the validity of the conclusion.

Affine periodic solutions for some stochastic differential equations

In this paper, we are study the problem of affine periodicity of solutions in distribution for some nonlinear stochastic differential equation with exponentially stable. We prove the existence and uniqueness of stochastic affine periodic solutions in distribution via the Banach fixed-point theorem.

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.

The Analytic solution for some non-linear stochastic differential equation by linearization (Linear-transform)

The Analytic solution for some non-linear stochastic differential equation by linearization (Linear-transform)

Closed-form and technology independent phase noise relation for LC oscillators

PurposeThe purpose of this paper is to find a closed relation for the phase noise of LC oscillators.Design/methodology/approachThe governing equation of oscillators is generally a stochastic nonlinear differential equation. In this paper, a closed relation for the phase noise of LC oscillators was obtained by approximating the I–V characteristic of the oscillator with third-degree polynomials and analyzing its differential equation.FindingsThis relation expresses phase noise directly in terms of circuit parameters, including the sizes of the transistors and the bias. Next, for evaluation, the phase noise of the cross-coupled oscillator without tail current was calculated with the proposed model. In this approach, the obtained equations are expressed independently of technology by combining the obtained phase noise relation and gm/ID method.Originality/valueA technology-independent method using the gm/ID method and the closed relationship is provided to calculate phase noise.

Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part M of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on “the future” of the process M as well as a dependence of its law (McKean–Vlasov-type equations). For such drivers, we prove the existence and uniqueness of a global solution (i.e. for any maturity T>0) to a BSDE with data satisfying natural integrability conditions.

A new hybrid approach for nonlinear stochastic differential equations driven by multifractional Gaussian noise

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems.