The composite Milstein methods for the numerical solution of Ito stochastic differential equations

Omar, Mohamed; Aboul-Hassan, Abdel-Karim; Rabia, Sherif I.

doi:10.1016/j.cam.2010.10.026

Cited by 23 publications

(14 citation statements)

References 13 publications

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“…The integral in equation (68) can be computed by using the composite Milstein method (MSII), introduced by Omar, Aboul-Hassan and Rabia [51]. This method is a numerical method for the strong solution of stochastic differential equations (SDE) that can be used to solve the following SDE driven by one Wiener process dX (t) = f (t, X (t)) dt + g (t, X (t)) dW (t) , where X (t) is a stochastic process, f (t, X (t)) is is the drift coefficient, g (t, X (t)) is the diffusion coefficient and W (t) is the standard Wiener process whose increment W (t) = W (t + t)−W (t) is a Gaussian random variable N(0, t).…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…Since the Wiener process, W (t), has infinite variation over every time interval, the last integral in equation (68) cannot be defined in the usual way (Riemann-Stieltjes or Lebesgue integral). This integral is of a stochastic type that can be solved in the sense of Itô calculus; see Kloeden and Platen [21], Klebaner [50], Lawler [22] and Omar, Aboul-Hassan and Rabia [51].…”

Section: Solution Under Stochastic Boundary Conditionsmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic thermal shock problem and study of wave propagation in the theory of generalized thermoelastic diffusion

Sherief

El‐Maghraby

Allam

2016

Mathematics and Mechanics of Solids

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A one-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. Two types of boundary conditions are considered, deterministic and stochastic. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. Analysis of wave propagation in the medium is presented. The solution in the physical domain for temperature, displacement, stress, concentration and chemical potential are obtained in an approximate manner. Numerical results are carried out and represented graphically.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Solution Under Stochastic Boundary Conditionsmentioning

confidence: 99%

Stochastic thermal shock problem and study of wave propagation in the theory of generalized thermoelastic diffusion

Sherief

El‐Maghraby

Allam

2016

Mathematics and Mechanics of Solids

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“…The family of semi-implicit (drift-implicit) methods has been known for a long time [11,25,33] and is well adapted to the problems with stiff deterministic part. For equations where both drift and diffusion parts are stiff, fully implicit schemes can be used at a cost of higher computational complexity [3,30,41]. Also, an elegant explicit approach based on a stochastic modification of Chebyshev methods which possess very good stability properties was recently proposed in [1,2].…”

Section: Introductionmentioning

confidence: 99%

Split-step Milstein methods for multi-channel stiff stochastic differential systems

Reshniak

Voss

Zhang

2015

Applied Numerical Mathematics

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We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for different noise and drift structures. The stability matrices are established in a form convenient for analyzing their impact arising from different deterministic drift integrators. Numerical examples are provided to illustrate the effectiveness and reliability of these methods.

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“…These integrals were found to be of a stochastic type, which are not solvable as a Riemann or Lebesgue integral. Stochastic integrals are mainly classified as Itô or Stratonovich integrals; see [30][31][32][33][34]. In our work, we consider the resultant integrals in Itô sense, which can be solved by many numerical methods; see [31,33,34].…”

Section: Introductionmentioning

confidence: 99%

“…Stochastic integrals are mainly classified as Itô or Stratonovich integrals; see [30][31][32][33][34]. In our work, we consider the resultant integrals in Itô sense, which can be solved by many numerical methods; see [31,33,34]. The objective of this study is to determine the effect of the stochastic bottom topography on the generation and propagation of the tsunami wave form and discuss aspects of tsunami generation that should be considered in developing this model as well as the propagation wave after the formation of the source model has been completed.…”

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Modeling of Tsunami Generation and Propagation under the Effect of Stochastic Seismic Fault Source Model in Linearized Shallow-Water Wave Theory

Allam

Omar

Ramadan

2014

ISRN Applied Mathematics

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Tsunami generation and propagation caused by stochastic seismic fault driven by two Gaussian white noises in the x- and y-directions are investigated. This model is used to study the tsunami amplitude amplification under the effect of the noise intensities, spreading uplift length and rise times of the three-dimensional stochastic fault source model. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space). The amplification of tsunami amplitudes builds up progressively as time increases during the generation process due to wave focusing while the maximum wave amplitude decreases with time during the propagation process due to the geometric spreading and also due to dispersion. The maximum amplitude amplification is proportional to the propagation length of the stochastic source model and inversely proportional to the water depth. The increase of the normalized noise intensities on the bottom topography leads to an increase in oscillations and amplitude in the free surface elevation. We derived and analyzed the mean and variance of the random tsunami waves as a function of the propagated uplift length, noise intensities, and the average depth of the ocean along the generation and propagation path.

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The composite Milstein methods for the numerical solution of Ito stochastic differential equations

Cited by 23 publications

References 13 publications

Stochastic thermal shock problem and study of wave propagation in the theory of generalized thermoelastic diffusion

Stochastic thermal shock problem and study of wave propagation in the theory of generalized thermoelastic diffusion

Split-step Milstein methods for multi-channel stiff stochastic differential systems

Three-Dimensional Modeling of Tsunami Generation and Propagation under the Effect of Stochastic Seismic Fault Source Model in Linearized Shallow-Water Wave Theory

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