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The solution of a parabolic stochastic partial differential equation (SPDE)driven by an infinite-dimensional Brownian motion is in general not asemi-martingale anymore and does in general not satisfy an It\^{o} formula likethe solution of a finite-dimensional stochastic ordinary differential equation(SODE). In particular, it is not possible to derive stochastic Taylorexpansions as for the solution of a SODE using an iterated application of theIt\^{o} formula. Consequently, until recently, only low order numericalapproximation results for such a SPDE have been available. Here, the fact thatthe solution of a SPDE driven by additive noise can be interpreted in the mildsense with integrals involving the exponential of the dominant linear operatorin the SPDE provides an alternative approach for deriving stochastic Taylorexpansions for the solution of such a SPDE. Essentially, the exponential factorhas a mollifying effect and ensures that all integrals take values in theHilbert space under consideration. The iteration of such integrals allows us toderive stochastic Taylor expansions of arbitrarily high order, which are robustin the sense that they also hold for other types of driving noise processessuch as fractional Brownian motion. Combinatorial concepts of trees and woodsprovide a compact formulation of the Taylor expansions.

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