Simulation and Approximation of Stochastic Processes by Spline Functions

A Local Linearization (LL) method for the numerical integration of Random Differential Equations (RDE) is introduced. The classical LL approach is adapted to this type of equations, which are defined by random vector fields that are typically at most Hölder continuous with respect to the time argument. The order of strong convergence of the method is studied. It turns out that the LL method improves the order of convergence of conventional numerical methods that have been applied to RDEs. Additionally, the performance of the LL method is illustrated by means of numerical simulations, which show that it behaves well even in those equations with complicated noisy dynamics where conventional methods fail. (2000): 34F05, 34K28, 60H25. AMS subject classification

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“…For instance, a natural approximation to the term ξ(t n + u) − ξ(t n ) is given by the following linear spline interpolation [23] …”

Section: Local Linearization Methodmentioning

confidence: 99%

The Local Linearization Method for Numerical Integration of Random Differential Equations

et al. 2005

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“…To start, a handy approximation to the term ξ(t n + u) − ξ(t n ) in φ should be chosen. For instance, the one given by the following linear spline interpolation [26] …”

Section: Local Linearization Schemesmentioning

confidence: 99%

Rate of convergence of Local Linearization schemes for random differential equations

Jiménez

Carbonell

2009

Bit Numer Math

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Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed, which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization. However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived for each particular scheme.

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“…1.4 Interpolation methods for stochastic processes Weba (1992) and Seleznjev (2000) use spline interpolation of random processes for several applications including, for example, simulating solutions of stochastic differential equations and optimization of designs. Weba (1992) restricts attention to cubic spline interpolation with Hermite-type conditions on the first derivative at the boundaries.…”

Section: The Parzen Estimatormentioning

confidence: 99%

“…Weba (1992) restricts attention to cubic spline interpolation with Hermite-type conditions on the first derivative at the boundaries. He claims, without providing details, that the method can be extended to other types of boundary conditions and to noncubic splines.…”

Section: The Parzen Estimatormentioning

confidence: 99%

Non parametric estimation of smooth stationary covariance functions by interpolation methods

Elogne

Perrin²,

Thomas‐Agnan

2007

Stat Infer Stoch Process

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Simulation and Approximation of Stochastic Processes by Spline Functions

Cited by 9 publications

References 19 publications

The Local Linearization Method for Numerical Integration of Random Differential Equations

The Local Linearization Method for Numerical Integration of Random Differential Equations

Rate of convergence of Local Linearization schemes for random differential equations

Non parametric estimation of smooth stationary covariance functions by interpolation methods

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