Improved Euler–Maruyama Method for Numerical Solution of the Itô Stochastic Differential Systems by Composite Previous-Current-Step Idea

Nouri, Kazem; Ranjbar, Hassan; Torkzadeh, Leila

doi:10.1007/s00009-018-1187-8

Cited by 15 publications

(5 citation statements)

References 19 publications

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“…Mathematical Problems in Engineering 3 (18) with initial (Y 1 (0), Y 2 (0)) � (6, 0.853). For analysis of the Davis-Skodje system (18), we choose the parameters as follows:…”

Section: Schemes Formulationmentioning

confidence: 99%

“…Stochastic ordinary di erential equations (SODEs) play a pivotal role in explaining some physical phenomena such as chemical reactions [9], nancial mathematics [10], mathematical ecology [11], epidemiology [12], medicine [13], and population dynamics [14]. Generally, SODEs cannot be solved analytical, but many numerical solutions can be found, for instance, the split-step theta Milstein method [15], the least-squares method [16], the discrete Temimi-Ansari method [17], the improved Euler-Maruyama method [18], the ve-stage Milstein method [19], the split-step Milstein method [20], the split-step Adams-Moulton Milstein method [21], the split-step forward Milstein method [22], and the Runge-Kutta method [23,24].…”

Section: Introductionmentioning

confidence: 99%

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Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

Ranjbar

Nouri

Torkzadeh

2022

Mathematical Problems in Engineering

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In the present investigation, new explicit approaches by the Milstein method and increment function of the Jacobian derivative of the drift coefficient are designed. Several numerical tests such as Cox–Ingersoll–Ross process, stochastic Brusselator, and Davis-Skodje system are presented to illustrate the accuracy and the efficiency of our schemes. Furthermore, we show that the strong convergence rate of our procedures is approximately one.

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“…Mathematical Problems in Engineering 3 (18) with initial (Y 1 (0), Y 2 (0)) � (6, 0.853). For analysis of the Davis-Skodje system (18), we choose the parameters as follows:…”

Section: Schemes Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

Ranjbar

Nouri

Torkzadeh

2022

Mathematical Problems in Engineering

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“…Apart from the computational inefficiency, the stochastic matrix exponential involve stochastic Brownian integrals which are not straightforward to evaluate and EM scheme is advantageous in such situations. Many efficient improvements in the EM method have been done for solving different types of SDEs such as composite EM [19], implicit Euler-Taylor [20], truncated EM method [21], composite previous-current-step EM [22]. Inspite of the modifications, these schemes are still approximate in nature.…”

Section: Introductionmentioning

confidence: 99%

A change of measure enhanced near exact Euler Maruyama scheme for the solution to nonlinear stochastic dynamical systems

Tripura¹,

Imran²,

Hazra³

et al. 2021

Preprint

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The present study utilizes the Girsanov transformation based framework for solving a nonlinear stochastic dynamical system in an efficient way in comparison to other available approximate methods. In this approach, a rejection sampling is formulated to evaluate the Radon-Nikodym derivative arising from the change of measure due to Girsanov transformation. The rejection sampling is applied on the Euler Maruyama approximated sample paths which draw exact paths independent of the diffusion dynamics of the underlying dynamical system. The efficacy of the proposed framework is ensured using more accurate numerical as well as exact nonlinear methods. Finally, nonlinear applied test problems are considered to confirm the theoretical results. The test problems demonstrates that the proposed exact formulation of the Euler-Maruyama provides an almost exact approximation to both the displacement and velocity states of a second order non-linear dynamical system. KeywordsStochastic differential equations • Euler Maruyama • change of measure • non-linear oscillator • stochastic exponential

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“…1 and Ch. 10] and in the articles [14][15][16][17][18], for example. Stochastic delay differential equations have also been successfully applied to model real problems in different settings.…”

Section: Introductionmentioning

confidence: 99%

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

2019

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In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay τ > 0: x (t) = ax(t) + bx(t − τ), t ≥ 0, with initial condition x(t) = g(t), −τ ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. By using L p-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an L p-solution too. An analysis of L p-convergence when the delay τ tends to 0 is also performed in detail.

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Improved Euler–Maruyama Method for Numerical Solution of the Itô Stochastic Differential Systems by Composite Previous-Current-Step Idea

Cited by 15 publications

References 19 publications

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

A change of measure enhanced near exact Euler Maruyama scheme for the solution to nonlinear stochastic dynamical systems

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

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