Taylor expansions of solutions of stochastic partial differential equations with additive noise
Abstract
The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinitedimensional Brownian motion is in general not a semimartingale anymore and does in general not satisfy an Itô formula like the solution of a finitedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 arXiv:
 arXiv:1010.0161
 Bibcode:
 2010arXiv1010.0161J
 Keywords:

 Mathematics  Probability
 EPrint:
 Published in at http://dx.doi.org/10.1214/09AOP500 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)