Towards a Systematic Linear Stability Analysis of Numerical Methods for Systems of Stochastic Differential Equations

Buckwar, Evelyn; Kelly, Cónall

doi:10.1137/090771843

Cited by 66 publications

(56 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It turns out that the stability behavior for the Euler-Maruyama method (or its generalization obtained by using the θ method for the drift term) applied to these more general test equations is essentially captured by the the linear test equation (2.6). Finally we mention that for non normal drift (2.6) can indeed fail to characterize the stability property (at least in the pre-asymptotic regime) of numerical methods [15,7]. This is already the case in the deterministic setting for the test equation y = λy (see [12,IV.11]).…”

Section: Stability Of Numerical Methodsmentioning

confidence: 93%

See 1 more Smart Citation

Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

Abdulle¹,

Blumenthal²

2013

Journal of Computational Physics

View full text Add to dashboard Cite

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods, the computational cost for stiff systems is significantly reduced, while keeping the computational algorithm fully explicit and easy to implement. Numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized MLMC method and corroborate our theoretical findings.

show abstract

Section: Stability Of Numerical Methodsmentioning

confidence: 93%

“…[26,23] but these studies do not allow for an easy characterization of stability criterion. Another attempt to generalize the linear test equation has been proposed in [7] using the theory of stochastic stabilization and destabilization [18]. Two sets of test equations with d = m = 2 and d = m = 3 have been considered.…”

Section: Stability Of Numerical Methodsmentioning

confidence: 99%

Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

Abdulle¹,

Blumenthal²

2013

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…This has been investigated in [33,29] but these studies do not allow for an easy characterization of stability criterion. Using the theory of stochastic stabilization and destabilization [22] an attempt to generalize the linear test equation has been proposed in [8], where two sets of test equations with N = m = 2 and N = m = 3 have been studied. The conclusion of these studies is that the stability behavior of the EulerMaruyama method (or its generalization obtained by using the θ method for the drift term) is essentially captured by the test equation (12).…”

Section: The Stochastic Scalar Test Equation With Multiplicative Noisementioning

confidence: 99%

“…The conclusion of these studies is that the stability behavior of the EulerMaruyama method (or its generalization obtained by using the θ method for the drift term) is essentially captured by the test equation (12). We mention however that for linear systems with a non normal drift, the additional test equations in [8] capture stability behaviors (in particular in the pre asymptotic regime) of a numerical scheme that cannot be seen by studying (12). This phenomenon is well known for ODEs (see [13,IV.11]).…”

Section: The Stochastic Scalar Test Equation With Multiplicative Noisementioning

confidence: 99%

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Abdulle¹,

Vilmart²,

Zygalakis³

2013

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

To cite this version:Assyr Abdulle, Gilles Vilmart, Konstantinos Zygalakis. Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2013, 35 (4) We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

show abstract

“…In particular, for nonlinear SDEs under the global Lipschitz condition, Higham, Mao, and Stuart [9] show that the exponential stability in mean square for the SDE is equivalent to the exponential stability in mean square of the numerical method (e.g., the EulerMaruyama and the stochastic theta method) for sufficiently small step sizes. For further developments in this area, we refer the reader to [5,8,19,22,24,27], for example, and the references therein.…”

mentioning

confidence: 99%

Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Mao¹

2015

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ∈ (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1.

show abstract

Towards a Systematic Linear Stability Analysis of Numerical Methods for Systems of Stochastic Differential Equations

Cited by 66 publications

References 12 publications

Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Contact Info

Product

Resources

About