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Zbl 1228.65015
Jentzen, Arnulf
Higher order pathwise numerical approximations of SPDEs with additive noise. (English)
[J] SIAM J. Numer. Anal. 49, No. 2, 642-667 (2011). ISSN 0036-1429; ISSN 1095-7170/e

The author considers the numerical approximation of semilinear parabolic stochastic partial differential equations (SPDEs) with additive noise in the form $$dX_{t}=[AX_{t}+F(X_{t})]dt+BdW_{t}.$$ In the equation, $X_{t}\ $ belongs to a Hilbert space $H,\ W_{t}$ is a Wiener process on a Hilbert space $U,\ A$ is an unbounded linear operator, $B$ is a bounded linear operator, $F$ is a nonlinear operator. The main result of the article shows that pathwise convergence of the method proposed by the author has a higher order convergence rate than the Euler scheme.
[Grigori N. Milstein (Yekaterinburg)]

MSC 2000:: *65C30 Stochastic differential and integral equations
60H15 Stochastic partial differential equations
35R60 PDE with randomness
35K58
65M12 Stability and convergence of numerical methods (IVP of PDE)

Keywords: stochastic partial differential equation; pathwise approximation; additive noise; fractional Brownian motion; Galerkin approximation; semilinear parabolic equations; Hilbert space; Wiener process; convergence

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Zentralblatt MATH Berlin [Germany]