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

TL;DR

This paper develops a method to derive high-order stochastic Taylor expansions for solutions of parabolic SPDEs with additive noise, using exponential integrals and combinatorial tree structures, overcoming limitations of traditional Itô calculus.

Contribution

It introduces a novel approach to obtain arbitrarily high-order stochastic Taylor expansions for SPDEs driven by additive noise, applicable to various noise types.

Findings

Derivation of high-order stochastic Taylor expansions for SPDEs.

Use of exponential integrals to ensure integrals take values in the Hilbert space.

Applicability to different noise processes like fractional Brownian motion.

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 It\^{o} 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 It\^{o} 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.