Euler-Maruyama Scheme Research Articles

On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient

De nombreux efforts ont été consacrés récemment à l’analyse de l’erreur $L_{p}$ de schéma d’Euler–Maruyama pour des équations différentielles stochastiques (EDS) avec un coefficient de dérive pouvant avoir des discontinuités en espace. Jusqu’à présent, pour des EDS scalaires avec un coefficient de dérive Lipschitz par morceaux et un coefficient de diffusion Lipschitz qui est non nul aux points de discontinuité du coefficient de dérive, seule une borne d’erreur $L_{p}$ avec un taux d’au moins $1/(2p)$ – a été obtenue. Dans cet article, nous montrons que sous les hypothèses précédentes, le schéma d’Euler–Maruyama réalise un taux d’erreur $L_{p}$ d’au moins $1/2$ pour tout $p\in [1,\infty )$, comme dans le cas d’EDS avec coefficients Lipschitz. La preuve de ce résultat se fonde sur une analyse détaillée de temps d’occupation bien choisis pour le schéma d’Euler–Maruyama.

Stochastic stability of a piezoelectric vibration energy harvester under a parametric excitation and noise-induced stabilization

Vibration energy harvesters seek to convert the energy of ambient, random vibrations into electrical power, often using piezoelectric transduction. The stochastic dynamics of a piezoelectric harvester subjected to a parametric (state-dependent) excitation has not been comprehensively investigated in the nonequilibrium regime. Motivated by mathematical results that establish the stabilization of dynamic response using noise, we investigate the stochastic stability of a generic harvester in the linear and the monostable nonlinear regimes excited by a multiplicative noise process characterized by both Brownian and Lévy distributions. Stability is characterized in each case by studying the approach of the harvester response towards equilibrium in the time domain, longer term proximity (or divergence) of two solutions starting with nearby initial conditions in the phase plane as well as the Lyapunov exponent. In the linear case, we analytically obtain a lower bound on the magnitude of the noise intensity that guarantees stability. The bound is derived as an inequality in terms of the system parameters. This analytic result is validated numerically using the Euler-Maruyama scheme by: (1) computing the harvester response and its approach to equilibrium in terms of its displacement, velocity and voltage, (2) computing the trajectories of two solutions with nearby initial conditions in the phase plane and (3) the sign of the maximal Lyapunov exponent in terms of energy. We find that noise-induced stabilization occurs consistently for the noise intensities greater than the lower bound in linear and weakly nonlinear harvesters, for both Brownian and Lévy excitation. Instabilities emerge for the noise intensities lower than the bound. The results lead to the interesting conclusion that noise of appropriate strength can induce stabilization and are expected to be useful in the design of energy harvesters. In addition, the results are expected to be significant in the study of phenomena such as noise-induced transport in Brownian rotors where stability is an important aspect of the stochastic dynamics.

Collective mode Brownian dynamics: A method for fast relaxation of statistical ensembles.

Sampling equilibrium configurations of correlated systems of particles with long relaxation times (e.g., polymeric solutions) using conventional molecular dynamics and Monte Carlo methods can be challenging. This is especially true for systems with complicated, extended bond network topologies and other interactions that make the use and design of specialized relaxation protocols infeasible. We introduce a method based on Brownian dynamics simulations that can reduce the computational time it takes to reach equilibrium and draw decorrelated samples. Importantly, the method is completely agnostic to the particle configuration and the specifics of interparticle forces. In particular, we develop a mobility matrix that excites non-local, collective motion of N particles and can be computed efficiently in O(N) time. Particle motion in this scheme is computed by integrating the overdamped Langevin equation with an Euler-Maruyama scheme, in which Brownian displacements are drawn efficiently using a low-rank representation of the mobility matrix in position and wave space. We demonstrate the efficacy of the method with various examples from the realm of soft condensed matter and release a massively parallel implementation of the code as a plugin for the open-source package HOOMD-blue [J. A. Anderson et al., J. Comput. Phys. 227, 5342 (2008) and J. Glaser et al., Comput. Phys. Commun. 192, 97 (2015)] which runs on graphics processing units.

Mean-Square Exponential Stability for Stochastic Control Systems With Discrete-Time State Feedbacks and Their Numerical Schemes in Simulation

Aiming at the qualitative numerical simulation, this article concerns the mean-square exponential stability (MSES) of the stochastic systems with discrete-time state feedbacks and their numerical schemes. A time delay dependent stability criterion is obtained for the underlying system. The Euler–Maruyama and backward Euler–Maruyama schemes are established, and their MSES are proved under the same criterion, which means that the numerical schemes preserve the stability of the continuous model. The illustrative example at the end of this letter shows that the theoretical upper bound for the sampling period limited in the stability criterion is much greater than that obtained by the stability criteria from the literature.

On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-H\"older drift in the recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

A Stochastic Model of Stress Evolution in a Bolted Structure in the Presence of a Joint Elastic Piece: Modeling and Parameter Inference

In this paper, we propose a stochastic model to describe over time the evolution of stress in a bolted mechanical structure depending on different thicknesses of a joint elastic piece. First, the studied structure and the experiment numerical simulation are presented. Next, we validate statistically our proposed stochastic model, and we use the maximum likelihood estimation method based on Euler–Maruyama scheme to estimate the parameters of this model. Thereafter, we use the estimated model to compare the stresses, the peak times, and extinction times for different thicknesses of the elastic piece. Some numerical simulations are carried out to illustrate different results.

Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment.

In the paper, we propose a novel stochastic population model with Markov chain and diffusion in a polluted environment. Under the condition that the diffusion coefficient satisfies the local Lipschitz condition, we prove the existence and uniqueness of invariant measure for the model. Moreover, we also discuss the existence and uniqueness of numerical invariance measure for stochastic population model under the discrete-time Euler-Maruyama scheme, and prove that numerical invariance measure converges to the invariance measure of the corresponding exact solution in the Wasserstein distance sense. Finally, we give the numerical simulation to show the correctness of the theoretical results.

Convergence, Non-negativity and Stability of a New Tamed Euler–Maruyama Scheme for Stochastic Differential Equations with Hölder Continuous Diffusion Coefficient

We propose and analyze a new tamed Euler–Maruyama approximation scheme for stochastic differential equations with Holder continuous diffusion. This new scheme preserves the stability and non-negativity of the exact solution.

Efficient discretisation of stochastic differential equations

The aim of this study is to find a generic method for generating a path of the solution of a given stochastic differential equation which is more efficient than the standard Euler–Maruyama scheme with Gaussian increments. First we characterize the asymptotic distribution of pathwise error in the Euler–Maruyama scheme with a general partition of time interval and then, show that the error is reduced by a factor when using a partition associated with the hitting times of sphere for the driving d-dimensional Brownian motion. This reduction ratio is the best possible in a symmetric class of partitions. Next we show that a reduction which is close to the best possible is achieved by using the hitting time of a moving sphere that is easier to implement.

Bayesian inference for a susceptible-exposed-infected-recovered epidemic model with data augmentation

ABSTRACT A Bayesian data-augmentation method allows estimating the parameters in a susceptible-exposed-infected-recovered (SEIR) epidemic model, which is formulated as a continuous-time Markov process and approximated by a diffusion process using the convergence of the master equation. The estimation was carried out with latent data points between every pair of observations simulated through the Euler-Maruyama scheme, which involves imputing the missing data in addition to the model parameters. The missing data and parameters are treated as random variables, and a Markov-chain Monte-Carlo algorithm updates the missing data and the parameter values. Numerical simulations show the effectiveness of the proposed Markov-chain Monte-Carlo algorithm.

Convergence rates of truncated theta-EM scheme for SDDEs

This paper is concerned with strong convergence of a truncated theta-EM (Euler-Maruyama) scheme for stochastic differential delay equations (SDDEs). Based on the truncated EM scheme, we generalize it to a truncated theta-EM scheme and discuss strong convergence rate for stochastic differential delay equations under local Lipschitz condition.

Strong convergence of the partially truncated Euler–Maruyama scheme for a stochastic age-structured SIR epidemic model

we study in this paper strong convergence of the partially truncated Euler-Maruyama scheme for an age-structured Susceptible-Infected-Removed (SIR) epidemic model with environmental noise. Using the semigroup theory, the existence and uniqueness of global positive solution for the model is first proved. We then define a truncated function and develop a partially truncated EM numerical solutions to the stochastic age-structured SIR epidemic model. We present the pth moment boundedness of the partially truncated EM numerical approximate solutions under appropriate conditions. Furthermore, the strong Lq convergence is established for the condition of 2 ≤ q < p of the partially truncated EM scheme. Finally, numerical simulations and examples are provided to demonstrate theoretical results and to illustrate validity of the partially truncated EM scheme.

A third-order weak approximation of multidimensional Itô stochastic differential equations

Abstract This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler–Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply implemented by a Monte Carlo method. The method of Watanabe distributions on Wiener space is effectively applied in the computation of the polynomial weight of Brownian motions. Numerical examples are shown to confirm the accuracy of the scheme.

Least Squares Estimator for Path-Dependent McKean-Vlasov SDEs via Discrete-Time Observations

In this article, we are interested in least squares estimator for a class of path-dependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown parameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of this article lie in three aspects: (i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no longer works; (ii) We take the advantage of linear interpolation with respect to the discrete-time observations to approximate the functional solution; (iii) Our model is more applicable and practice as we are dealing with SDEs with irregular coefficients (for example, Holder continuous) and path-distribution dependent.

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.

On the Euler–Maruyama scheme for SDEs with bounded variation and Hölder continuous coefficients

We consider the strong rate of convergence of the Euler–Maruyama approximation for stochastic differential equations with possibly discontinuous drift and Hölder continuous diffusion coefficient. In particular, we show that the rates obtained in some recent papers can be improved under an additional assumption that the diffusion coefficient is of bounded variation.

Numerical solution to highly nonlinear neutral-type stochastic differential equation

In the paper, our main aim is to investigate the strong convergence of the implicit numerical approximations for neutral-type stochastic differential equations with super-linearly growing coefficients. After providing mean-square moment boundedness and mean-square exponential stability for the exact solution, we show that a backward Euler–Maruyama approximation converges strongly to the true solution under polynomial growth conditions for sufficiently small step size. Imposing a few additional conditions, we examine the p-th moment uniform boundedness of the exact and approximate solutions by the stopping time technique, and establish the convergence rate of one half, which is the same as the convergence rate of the classical Euler–Maruyama scheme. Finally, several numerical simulations illustrate our main results.

Operator Splitting Around Euler-Maruyama Scheme and High Order Discretization of Heat Kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

An Adaptive Euler--Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis

We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an adaptive step sizing strategy for the explicit Euler-Maruyama scheme. As a result, we obtain a numerical method which has -- up to logarithmic terms -- strong convergence order $1/2$ with respect to the average computational cost. We support our theoretical findings with several numerical examples.

Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation

In this paper, we describe the two different stochastic differential equations representing cholera dynamics. The first stochastic differential equation is formulated by introducing the stochasticity to deterministic model by parametric perturbation technique which is a standard technique in stochastic modeling and the second stochastic differential equation is formulated using transition probabilities. We analyse a stochastic model using suitable Lyapunov function and Ito formula. We state and prove the conditions for global existence, uniqueness of positive solutions, stochastic boundedness, global stability in probability, moment exponential stability, and almost sure convergence. We also carry out numerical simulation using Euler-Maruyama scheme to simulate the sample paths of stochastic differential equations. Our results show that the sample paths are continuous but not differentiable (a property of Wiener process). Also, we compare the numerical simulation results for deterministic and stochastic models. We find that the sample path of SIsIaR-B stochastic differential equations model fluctuates within the solution of the SIsIaR-B ordinary differential equation model. Furthermore, we use extended Kalman filter to estimate the model compartments (states), we find that the state estimates fit the measurements. Maximum likelihood estimation method for estimating the model parameters is also discussed.