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Time-discretised Galerkin approximations of parabolic stochastic PDE's

Published online by Cambridge University Press:  17 April 2009

W. Grecksch
Affiliation:
Institut für Optimierung und Stochastik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Hale (Saale), Germany.
P.E. Kloeden
Affiliation:
School of Computing and Mathematics, Deakin University, Geelong Campus, Geelong Vic. 3217, Australia.
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Abstract

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The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above by

where [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Prato, G. Da and Zabczyk, G., Stochastic equations in infinite dimensions (Cambridge University Press, Cambridge, 1992).Google Scholar
[2]Grecksch, W. and Tudor, C., Stochastic evolution equations (Akademie—Verlag, Berlin, 1995).Google Scholar
[3]Kloeden, P.E. and Platen, E., Numerical solution of stochastic differential equations (Springer-Verlag, Berlin, Heidelberg, New York, 1992).Google Scholar