The mean-square stability of numerical methods for solving stochastic differential equations

Artem'ev, S. S.

doi:10.1515/rnam.1994.9.5.405

Cited by 3 publications

(3 citation statements)

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“…(Note that the 1-stage drift-implicit Rosenbrock method applied to bilinear systems (1.2) coincides with that of drift-implicit Theta methods with choice a D Â.) Both [3], [4] and [27] exploit the relation of asymptotic stability to the positivedefinite solvability of Lyapunov equation (i.e. the existence of positive-definite S for positive-definite matrix C )…”

Section: Appendix: a Comment On Stability Investigations Of Stochastimentioning

confidence: 96%

“…bilinear systems, linear in drift and diffusion Brought to you by | University of Manitoba Authenticated Download Date | 6/12/15 12:27 PM terms). A first approach to multi-dimensional setting of linear stochastic numerical systems is found in Artemiev [2], [3], Artemiev and Averina [4], Korzeniowski [13] and Schurz [27], [29], [30], [31]. The stability analysis in Artemiev and Averina [4] is motivated from the usage of stochastic Rosenbrock-type methods.…”

Section: Appendix: a Comment On Stability Investigations Of Stochastimentioning

confidence: 99%

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Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in

Schurz

2011

Random Operators and Stochastic Equations

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Almost sure asymptotic stability of trivial solution and almost sure convergence of stochastic Theta methods applied to bilinear systems of ordinary stochastic differential equations (SDEs) of Itô-type in R d are proven. For this purpose, we prove and exploit a convergence theorem for non-negative semi-martingale decompositions, and verify a practical criteria based on the uniform boundedness of nonrandom eigenvalues related to certain matrix systems in any dimension d . We do not assume commutativity or simultaneous diagonalizability of drift and diffusion parts as many other authors, neither we restrict our analysis and applicability to only 2D or 3D cases nor to uniform step sizes (since the problem of adequate stochastic test equations cannot be solved within nonanticipative Itô calculus). However, an example of 2D diagonal-noised systems illustrates our approach. The discrete time systems of stochastic Theta methods are driven by L 2martingales (i.e. martingale differences, not necessarily Gaussian) and can be interpreted as nonautonomous discretizations (e.g. with variable step sizes or dependence on time).

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Section: Appendix: a Comment On Stability Investigations Of Stochastimentioning

confidence: 96%

Section: Appendix: a Comment On Stability Investigations Of Stochastimentioning

confidence: 99%

Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in

Schurz

2011

Random Operators and Stochastic Equations

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“…This class of test problems is a natural analogue to the linear test problems used to analyze stability properties (such as A-stability) of numerical schemes for ordinary differential equations (ODEs) and has been considered previously by several authors [14,19,27,26,23,9,7,16,15,5] for schemes using a fixed time-step h. Linear test problems are of interest because complete analyses are often possible which, via linearization arguments, can provide insights into the behaviour of numerical schemes on more general classes of problem. Other investigations of mean-square stability with fixed time-steps include [2,3,22,28,1,24,29,8,30].…”

Section: Introductionmentioning

confidence: 99%

Mean-square stability properties of an adaptive time-stepping SDE solver

Lamba

Seaman

2006

Journal of Computational and Applied Mathematics

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We consider stability properties of a class of adaptive time-stepping schemes based upon the Milstein method for stochastic differential equations with a single scalar forcing. In particular we focus upon mean-square stability for a class of linear test problems with multiplicative noise. We demonstrate that highly desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls, one for the drift term and one for the diffusion.

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A brief review on stability investigations of numerical methods for systems of stochastic differential equations

Schurz

2024

NHM

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<abstract><p>A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.</p></abstract>

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The mean-square stability of numerical methods for solving stochastic differential equations

Cited by 3 publications

References 4 publications

Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in

Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in

Mean-square stability properties of an adaptive time-stepping SDE solver

A brief review on stability investigations of numerical methods for systems of stochastic differential equations

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