The Annals of Probability
- Ann. Probab.
- Volume 38, Number 2 (2010), 532-569.
Taylor expansions of solutions of stochastic partial differential equations with additive noise
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.
Ann. Probab. Volume 38, Number 2 (2010), 532-569.
First available in Project Euclid: 9 March 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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.