A Theorem on the Order of Convergence of Mean-Square Approximations of Solutions of Systems of Stochastic Differential Equations

Abstract.A new approach to the construction of mean-square numerical methods for the solution of stochastic differential equations with small noises is proposed. The approach is based on expanding the exact solution of the system with small noises in powers of time increment and small parameter. The theorem on the mean-square estimate of method errors is proved. Various efficient numerical schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The proposed methods are tested by calculation of Lyapunov exponents and simulation of a laser Langevin equation with multiplicative noises.

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“…The proof of Theorem 3.1 is similar to the proof of the mean-square convergence theorem for a general system [6], [8], and here it is omitted.…”

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confidence: 99%

Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises

Milstein¹,

Tretyakov²

1997

SIAM J. Sci. Comput.

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“…Here the constant C is again independent of h. For the strong convergence we have the following result [26]. If the functions f, g r are sufficiently smooth and Lipschitz continuous and…”

Section: Weak Stochastic Integratorsmentioning

confidence: 94%

“…We refer to the proof of [4, Thm. 4.2] for details on deriving the estimates (26) in the context of stochastic stabilized explicit integrators.…”

Section: With (41)mentioning

confidence: 99%

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Abdulle¹,

Vilmart²,

Zygalakis³

2013

SIAM J. Sci. Comput.

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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.

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“…In particular, implicit or predictor-corrector methods are used to control the propagation of errors. We refer for this approach to papers by [1], [2], [7], [8], [11], [12], [15], [16], [20], [21] and [22]. There are various numerical schemes that perform well on some SDEs for certain parameter ranges and sufficiently small step sizes.…”

Section: Introductionmentioning

confidence: 99%

Quasi-exact approximation of hidden Markov chain filters

Platen

Rendek

2010

COSA

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Abstract. This paper studies the application of exact simulation methods for multi-dimensional multiplicative noise stochastic differential equations to filtering. Stochastic differential equations with multiplicative noise naturally occur as Zakai equation in hidden Markov chain filtering. The paper proposes a quasi-exact approximation method for hidden Markov chain filters, which can be applied when discrete time approximations, such as the Euler scheme, may fail in practice.

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A Theorem on the Order of Convergence of Mean-Square Approximations of Solutions of Systems of Stochastic Differential Equations

Cited by 47 publications

References 2 publications

Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises

Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noises

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Quasi-exact approximation of hidden Markov chain filters

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