The Annals of Probability

Taylor expansions of solutions of stochastic partial differential equations with additive noise

Arnulf Jentzen and Peter Kloeden

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Abstract

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itô formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Itô formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

Article information

Source
Ann. Probab. Volume 38, Number 2 (2010), 532-569.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143526

Digital Object Identifier
doi:10.1214/09-AOP500

Mathematical Reviews number (MathSciNet)
MR2642885

Zentralblatt MATH identifier
1220.35202

Subjects
Primary: 35K90: Abstract parabolic equations 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 65C30: Stochastic differential and integral equations 65M99: None of the above, but in this section

Keywords
Taylor expansions stochastic partial differential equations SPDE strong convergence stochastic trees

Citation

Jentzen, Arnulf; Kloeden, Peter. Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Probab. 38 (2010), no. 2, 532--569. doi:10.1214/09-AOP500. https://projecteuclid.org/euclid.aop/1268143526.


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