The Local Linearization Method for Numerical Integration of Random Differential Equations

Carbonell, Félix; Jiménez, Juan Carlos; Biscay, Rolando J.; Cruz, H. de la

doi:10.1007/s10543-005-2645-9

Cited by 32 publications

(42 citation statements)

References 20 publications

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“…The local linearization method (LL) for RODEs was proposed by Carbonell et al [3], where its discretization error was analyzed. In our context the LL scheme has the form…”

Section: The Local Linearization Scheme For Rodesmentioning

confidence: 99%

Pathwise Taylor schemes for random ordinary differential equations

Jentzen

Kloeden

2009

Bit Numer Math

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Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.

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“…The local linearization method (LL) for RODEs was proposed by Carbonell et al [3], where its discretization error was analyzed. In our context the LL scheme has the form…”

Section: The Local Linearization Scheme For Rodesmentioning

confidence: 99%

Pathwise Taylor schemes for random ordinary differential equations

Jentzen

Kloeden

2009

Bit Numer Math

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“…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 proposed in [3] and with a suitable computational effort.…”

Section: Introductionmentioning

confidence: 99%

“…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

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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

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“…Adicionalmente, permite integrar ciertos tipos de ecuaciones "stiff" con menor costo computacional y mejor precisión que los integradores explícitos convencionales. Otra característica ventajosa del método LL es la flexibilidad de su aplicación a otras clases de ecuaciones diferenciales más complejas como por ejemplo las ecuaciones diferenciales estocásticas [8,2], las aleatorias [3] y las ecuaciones con retardo [17].…”

Section: Introductionunclassified