Pathwise Taylor schemes for random ordinary differential equations

Jentzen, Arnulf; Kloeden, Peter E.

doi:10.1007/s10543-009-0211-6

Cited by 25 publications

(14 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the basis of Taylor expansion (11), the growth condition (4) and the assumptions A 1 , A 2 and A 3 , we conclude that J 1 (t) can be estimated in the following way,…”

Section: Resultsmentioning

confidence: 91%

“…(1) and the approximate solution x n decreases when the degrees of Taylor expansions for f and g increase, indicates that it would be convenient to combine the presented analytic method with numerical approximations based on Ito-Taylor expansions of higher degrees, described, above all, by Kloeden and Platen [8,9]. Moreover, in order to derive numerical schemes of higher order, it seems to be reasonable to replace the solutions in approximate equations, that is, in Taylor polynomials, by lower order approximations, analogously to the recent papers [10,11] by Kloeden and Jentzen treating random ordinary differential equations.…”

mentioning

confidence: 80%

See 1 more Smart Citation

An approximate method via Taylor series for stochastic functional differential equations

Milošević

Jovanović

Janković

2010

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the L p -norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.

show abstract

“…On the basis of Taylor expansion (11), the growth condition (4) and the assumptions A 1 , A 2 and A 3 , we conclude that J 1 (t) can be estimated in the following way,…”

Section: Resultsmentioning

confidence: 91%

mentioning

confidence: 80%

An approximate method via Taylor series for stochastic functional differential equations

Milošević

Jovanović

Janković

2010

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…In this connection it is important to know that due to a lack of time differentiability of the noise process, this method is in general not attaining its usual order of convergence. Details on specific integration schemes for random differential equations can be looked up in the publications [19] and [30].…”

Section: Applicationmentioning

confidence: 99%

Exploring the qualitative behavior of uncertain dynamical systems

Gaull

Kreuzer

2010

Nonlinear Dyn

View full text Add to dashboard Cite

The global behavior of uncertain nonlinear systems by means of the cell mapping method is investigated. The systems under consideration are governed by ordinary differential equations where uncertainty is characterized by a bounded stationary noise process. In order to make cell mapping methods applicable to this particular situation generalizations of the theory become necessary. The numerical approach is based on a reformulation of the dynamics in terms of a Markov process whose qualitative behavior is analyzed. We demonstrate the usefulness of the presented scheme in the engineering field while addressing an example problem from technical mechanics.

show abstract

“…In particular high accuracy, computational efficiency, and stability of the numerical schemes, are very desirable properties. Taking all this into consideration, some numerical integrator have been proposed in literature e.g., [2], [6], [8], [3]. However, these methods or are of implicit nature (involving the numerical solution of a system of nonlinear algebraic equations at each integration step, that typically increase the computational effort of these numerical integrators) or are explicit integrators, having the appealing feature of retaining the standard order of convergence of the classical deterministic schemes, but at the expense of high computational cost and low stability.…”

Section: Introductionmentioning

confidence: 99%

A higher order and stable method for the numerical integration of Random Differential Equations

Cruz

Jiménez²

2015

Anais Do Congresso Nacional De Matemática Aplicada À Indústria

View full text Add to dashboard Cite

Abstract. Over the last few years there has been a growing and renovated interest in the numerical study of Random Differential Equations (RDEs). On one hand it is motivated by the fact that RDEs have played an important role in the modeling of physical, biological, neurological and engineering phenomena, and on the other hand motivated by the usefulness of RDEs for the numerical analysis of Ito-stochastic differential equations (SDEs) -via the extant conjugacy property between RDEs and SDEs-, which allows to study stronger pathwise properties of SDEs driven by different kind of noises others than the Brownian. Since in most common cases no explicit solution of the equations is known, the construction of computational methods for the treatment and simulation of RDEs has become an important need. In this direction the Local Linearization (LL) approach is a successful technique that has been applied for defining numerical integrators for RDEs. However, a major drawback of the obtained methods is its relative low order of convergence; in fact it is only twice the order of the moduli of continuity of the driven stochastic process. The present work overcomes this limitation by introducing a new, exponential-based, high order and stable numerical integrator for RDEs. For this, a suitable approximation of the stochastic processes present in the random equation, together with the local linearization technique and an adapted Padé method with scaling and squaring strategy are conveniently combined. In this way a higher order of convergence can be achieved (independent of the moduli of continuity of the stochastic processes) while retaining the dynamical and numerical stability properties of the low order LL methods. Results on the convergence and stability of the suggested method and details on its efficient implementation are discussed. The performance of the introduced method is illustrated through computer simulations.Palavras-chave: random differential equations, numerical integrators, exponential methods, Local linearization methods

show abstract

Pathwise Taylor schemes for random ordinary differential equations

Cited by 25 publications

References 13 publications

An approximate method via Taylor series for stochastic functional differential equations

An approximate method via Taylor series for stochastic functional differential equations

Exploring the qualitative behavior of uncertain dynamical systems

A higher order and stable method for the numerical integration of Random Differential Equations

Contact Info

Product

Resources

About