A Lax equivalence theorem for stochastic differential equations

Lang, Annika

doi:10.1016/j.cam.2010.05.001

Cited by 10 publications

(3 citation statements)

References 10 publications

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“…To solve the stochastic system (2.11), we extend the SSPRK3 scheme used in the deterministic case. We will show that the numerical scheme we introduce is consistent in the sense of Definition 3, see Lang [2010].…”

Section: Numerical Implementationmentioning

confidence: 97%

Numerically Modeling Stochastic Lie Transport in Fluid Dynamics

Cotter¹,

Crisan²,

Holm³

et al. 2019

Multiscale Model. Simul.

100

203

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We present a numerical investigation of stochastic transport in ideal fluids. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. In the current paper, we develop new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation for incompressible 2D Euler fluid flows. The new methodology tested here is found to be suitable for coarse graining in this case. Specifically, we perform uncertainty quantification tests of the velocity decomposition of Cotter et al. (2017), by comparing ensembles of coarse-grid realisations of solutions of the resulting stochastic partial differential equation with the "true solutions" of the deterministic fluid partial differential equation, computed on a refined grid. The time discretisation used for approximating the solution of the stochastic partial differential equation is shown to be consistent. We include comprehensive numerical tests that confirm the non-Gaussianity of the stream function, velocity and vorticity fields in the case of incompressible 2D Euler fluid flows.

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Section: Numerical Implementationmentioning

confidence: 97%

Numerically Modeling Stochastic Lie Transport in Fluid Dynamics

Cotter¹,

Crisan²,

Holm³

et al. 2019

Multiscale Model. Simul.

100

203

View full text Add to dashboard Cite

show abstract

“…The Lax Equivalence Theorem states the necessary and sufficient condition for convergence. For a consistent finite difference scheme, stability is equivalent to convergence [38].…”

Section: Convergence Of the Fwp Predictive Modelmentioning

confidence: 99%

Heavy Tail and Long-Range Dependence for Skewed Time Series Prediction Based on a Fractional Weibull Process

Song,

Chen,

Zio

2023

Fractal Fract

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In this paper, a fractional Weibull process is utilized in a predictive stochastic differential equation model to allow for skewness and heavy-tailed characteristics. To this aim, a fractional Weibull process with non-Gaussian characteristics and a long memory effect is proposed to drive the predictive stochastic differential equation. The difference iterative forecasting model is proposed as its stochastic difference scheme. The consistency, stability, and convergence of the model are analyzed. In the proposed model, variational mode decomposition is utilized as the data preprocessing approach to separate the stationary and non-stationary components. Actual wind speed data and stock price data are employed in two separate case studies.

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“…Let h, ∆t > 0 be fixed. The family (D stoch,j ∆t,h , j ∈ N 0 ) is F-compatible in the sense of [12,21], i.e., D stoch,j ∆t,h is F t j+1 -measurable and E[D stoch,j ∆t,h |F t j ] = 0 for all j ∈ N 0 . Furthermore, for all j ∈ N 0 , let D stoch,j ∆t,h…”

Section: Asymptotic Mean-square Stability Analysismentioning

confidence: 99%