Taylor expansions of solutions of stochastic partial differential equations

TL;DR

This paper develops a novel method to derive high-order Taylor expansions for solutions of stochastic partial differential equations (SPDEs) without relying on the Itô formula, using recursive classical Taylor expansions and combinatorial structures.

Contribution

It introduces an alternative approach to obtain stochastic Taylor expansions for SPDEs, bypassing the limitations of Itô calculus for infinite-dimensional martingale-driven equations.

Findings

Derived high-order stochastic Taylor expansions for SPDE solutions.

Used combinatorial trees and woods for compact formulation.

Provided a recursive method avoiding Itô formula limitations.

Abstract

The solutions of parabolic and hyperbolic stochastic partial differential equations (SPDEs) driven by an infinite dimensional Brownian motion, which is a martingale, are in general not semi-martingales any more and therefore do not satisfy an It\^o formula like the solutions of finite dimensional stochastic differential equations (SODEs). In particular, it is not possible to derive stochastic Taylor expansions as for the solutions of SODEs using an iterated application of the It\^o formula. However, in this article we introduce Taylor expansions of solutions of SPDEs via an alternative approach, which avoids the need of an It\^o formula. The main idea behind these Taylor expansions is to use first classical Taylor expansions for the nonlinear coefficients of the SPDE and then to insert recursively the mild presentation of the solution of the SPDE. The iteration of this idea allows us to derive stochastic Taylor expansions of arbitrarily high order. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.