A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag

We introduce a modified Milstein scheme for pathwise approximation of scalar stochastic delay differential equations with constant time lag on a fixed finite time interval. Our algorithm is based on equidistant evaluation of the driving Brownian motion and is simply obtained by replacing iterated It...

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Veröffentlicht in:Journal of computational and applied mathematics Jg. 197; H. 1; S. 89 - 121
Hauptverfasser: Hofmann, Norbert, Müller-Gronbach, Thomas
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 01.12.2006
Elsevier
Schlagworte:
Asymptotic optimality
Exact error formulas
Exact sciences and technology
Mathematics
Minimal errors
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in probability and statistics
Pathwise approximation
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Stochastic analysis
Stochastic delay differential equations
Exact error formulas
secondary: 60H10
Asymptotic optimality
primary: 65C30
Stochastic delay differential equations
Pathwise approximation
Minimal errors
Error estimation
Differential equation
Minimal error
Delay equation
Euler scheme
Linear test
Numerical approximation
Brownian motion
Numerical analysis
Applied mathematics
primary: 65C30; secondary: 60H10
Exact error formula
Integral equation
Stochastic delay differential equations; Pathwise approximation; Asymptotic optimality; Minimal errors; Exact error formulas
Stochastic equation
ISSN:0377-0427, 1879-1778
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Beschreibung
Zusammenfassung:We introduce a modified Milstein scheme for pathwise approximation of scalar stochastic delay differential equations with constant time lag on a fixed finite time interval. Our algorithm is based on equidistant evaluation of the driving Brownian motion and is simply obtained by replacing iterated Itô-integrals by products of appropriate Brownian increments in the definition of the Milstein scheme. We prove that the piecewise linear interpolation of the modified Milstein scheme is asymptotically optimal with respect to the mean square L 2 -error within the class of all pathwise approximations that use observations of the driving Brownian motion at equidistant points. Moreover, for a large class of equations our scheme is also asymptotically optimal for mean square approximation of the solution at the final time point. Our asymptotic optimality results are complemented by a comparison with the Euler scheme based on exact error formulas for a linear test equation. This comparison demonstrates the superiority of the modified Milstein scheme even for a very small number of discretization points. Finally, we provide a generalization of our approach to the case of a system of SDDEs with an arbitrary finite number of constant delays. We conjecture that the above optimality results carry over to the generalized scheme.
Bibliographie:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2005.10.027