Euler-Maruyama Scheme Research Articles

On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.

Positivity and Boundedness Preserving Numerical Scheme for a Stochastic Multigroup Susceptible-Infected-Recovering Epidemic Model with Age Structure.

Since the stochastic age-structured multigroup susceptible-infected-recovering (SIR) epidemic model is nonlinear, the solution of this model is hard to be explicitly represented. It is necessary to construct effective numerical methods so as to predict the number of infections. In addition, the stochastic age-structured multigroup SIR model has features of positivity and boundedness of the solution. Therefore, in this article, in order to ensure that the numerical and analytical solutions must have the same properties, by modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving EM (PBPEM) method on temporal space for stochastic age-structured multigroup SIR model, which is proved to have a strong convergence to the true solution over finite time intervals. Moreover, by combining the standard finite element method and the PBPEM method, we propose a full-discrete scheme to show the numerical solutions, as well as analyze the error estimations. Finally, the full-discrete scheme is applied to a general stochastic two-group SIR model and the Chlamydia epidemic model, which shows the superiority of the numerical method.

Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel

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.

Insights of infected Schwann cells extinction and inherited randomness in a stochastic model of leprosy

Investigating disease progression, transmission of infection and impacts of Multidrug Therapy (MDT) to inhibit demyelination in leprosy involves a certain amount of difficulty in terms of the in-built uncertain complicated and complex intracellular cell dynamical interactions. To tackle this scenario and to elucidate a more realistic, rationalistic approach of examining the infection mechanism and associated drug therapeutic interventions, we propose a four-dimensional ordinary differential equation-based model. Stochastic processes has been employed on this deterministic system by formulating the Kolmogorov forward equation introducing a transition state and the quasi-stationary distribution, exact distribution analysis have been investigated which allow us to estimate an expected time to extinction of the infected Schwann cells into the human body more prominently. Additionally, to explore the impact of uncertainty in the key intracellular factors, the stochastic system is investigated incorporating random perturbations and environmental noises in the disease dissemination, proliferation and reinfection rates. Rigorous numerical simulations validating the analytical outcomes provide us significant novel insights on the progression of leprosy and unravelling the existing major treatment complexities. Analytical experiments along with the simulations utilizing Monte-Carlo method and Euler–Maruyama scheme involving stochasticity predicts that the bacterial density is underestimated due to the recurrence of infection and suggests that maintaining a drug-efficacy rate in the range 0.6−0.8 would be substantially efficacious in eradicating leprosy.

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

A fast Euler–Maruyama scheme and its strong convergence for multi-term Caputo tempered fractional stochastic differential equations

In this paper, we consider a kind of multi-term Caputo tempered fractional stochastic differential equations and prove the existence and uniqueness of the true solution. Then we derive an Euler–Maruyama (EM) scheme to solve the considered equations. In view of the huge computational cost caused by the EM scheme to achieve reasonable accuracy, a fast EM scheme is proposed based on the sum-of-exponentials approximation to improve its computational efficiency. Moreover, the strong convergence of our two numerical schemes are proved. Finally, several numerical examples are carried out to support our theoretical results and demonstrate the superior computational efficiency of the fast EM scheme.

Euler–Maruyama scheme for SDE driven by Lévy process with Hölder drift

This study focuses on approximating solutions to SDEs driven by Lévy processes with Hölder continuous drifts using the Euler–Maruyama scheme. We derive the Lp-error for a broad range of driven noises, including all nondegenerate α-stable processes (0<α<2).

Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.

On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

Normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for the SDE

Normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme for the SDE

On the weak rate of convergence for the Euler–Maruyama scheme with Hölder drift

We consider stochastic differential equations with bounded α-Hölder continuous drift, with α∈(0,1), driven by multiplicative noise. We moreover assume that the diffusion coefficient is uniformly elliptic and bounded, with bounded first and second order partial derivatives. We show that the weak rate of convergence for the Euler–Maruyama scheme is arbitrarily close to (1+α)/2.

An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of [Formula: see text] queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of [Formula: see text] queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role. Funding: X. Jin is supported in part by the Fundamental Research Funds for the Central Universities [Grants JZ2022HGQA0148 and JZ2023HGTA0170]. G. Pang is supported in part by the U.S. National Science Foundation [Grants DMS-1715875 and DMS-2216765]. L. Xu is supported in part by the National Nature Science Foundation of China [Grant 12071499], Macao Special Administrative Region [Grant FDCT 0090/2019/A2], and the University of Macau [Grant MYRG2018-00133-FST]. This work was supported by U.S. National Science Foundation [Grant DMS-2108683].

Convergence of Relative Entropy for Euler-Maruyama Scheme to Stochastic Differential Equations with Additive Noise.

For a family of stochastic differential equations driven by additive Gaussian noise, we study the asymptotic behaviors of its corresponding Euler-Maruyama scheme by deriving its convergence rate in terms of relative entropy. Our results for the convergence rate in terms of relative entropy complement the conventional ones in the strong and weak sense and induce some other properties of the Euler-Maruyama scheme. For example, the convergence in terms of the total variation distance can be implied by Pinsker's inequality directly. Moreover, when the drift is β(0<β<1)-Hölder continuous in the spatial variable, the convergence rate in terms of the weighted variation distance is also established. Both of these convergence results do not seem to be directly obtained from any other convergence results of the Euler-Maruyama scheme. The main tool this paper relies on is the Girsanov transform.

Strong convergence of Euler–Maruyama schemes for doubly perturbed McKean–Vlasov stochastic differential equations

In this paper, we develop strong convergence of the Euler–Maruyama (EM) scheme for approximating the doubly perturbed McKean–Vlasov stochastic differential equations. In contrast to the existing work, a novel feature is that we use more general conditions for parameters α and β. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean–Vlasov SDEs. Combining with the results of propagation of chaos, the overall convergence rate is obtained for the EM scheme. Finally, two numerical examples are provided to demonstrate our results.

Convergence rate of the Euler–Maruyama scheme applied to diffusion processes with Lq−Lρ drift coefficient and additive noise

Convergence rate of the Euler–Maruyama scheme applied to diffusion processes with Lq−Lρ drift coefficient and additive noise

Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

&lt;abstract&gt;&lt;p&gt;In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.&lt;/p&gt;&lt;/abstract&gt;

Splitting for some classes of homeomorphic and coalescing stochastic flows

The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E. Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.

Novel intelligent predictive networks for analysis of chaos in stochastic differential SIS epidemic model with vaccination impact

In this paper, stochastic predictive computing networks are exploited to investigate the dynamics of the SIS with vaccination impact based epidemic model (SISV-EM) represented by nonlinear systems of stochastic differential equations (SDEs) by exploitation of artificial neural networks (ANNs) with the backpropagated Levenberg-Marquardt technique (BLMT) i.e., (ANNs-BLMT) to approximate the solution behavior. The stochastic nonlinear SISV-EM is governed with three classes: susceptible, infectious, and vaccinated populations. The referenced or target datasets for ANNs-BLMT are constructed by employing Euler-Maruyama (EM) scheme for solving stochastic differential systems in case of sufficiently various nonlinear SISV-EM scenarios by varying the percentage of vaccination for newly born, the coefficient of transmission, the natural mortality rates, the infectious rates of recovery, the rate at which vaccinated people lose their immunity, the rate of death caused by disease, the proportion of vaccinated against susceptible and the white noise in the environment. Based on arbitrary training, testing, and validation samples from the referenced dataset, the ANNs-BLMT provides an approximate solution for the stochastic nonlinear SISV-EM, with significant correlations to the referenced results. Exhaustive simulation-based results using error histograms, mean square errors, and regression analyses further demonstrate that the proposed ANNs-BLMT is efficient, consistent, and accurate for solving SISV-EM.

Convergence rate in [formula omitted] sense of tamed EM scheme for highly nonlinear neutral multiple-delay stochastic McKean–Vlasov equations

This paper focuses on the numerical scheme of highly nonlinear neutral multiple-delay stochastic McKean-Vlasov equation (NMSMVE) by virtue of the stochastic particle method. First, under general assumptions, the results about propagation of chaos in Lp sense are revealed, where the convergence rate loses a little due to the proof technique. Then the tamed Euler–Maruyama scheme to the corresponding particle system is established and the convergence rate in Lp sense is obtained. Furthermore, combining these two results gives the convergence error in Lp sense between the objective NMSMVE and numerical approximation, which is related to the particle number and step size. Finally, two numerical examples are provided to support the finding.

Truncated Euler–Maruyama method for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient

In this paper, we propose a truncated Euler–Maruyama scheme for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient. Meanwhile, the convergence rate of the numerical method is established. Numerical example is demonstrated to verify the theoretical results.