Solving SPDEs driven by colored noise: a chaos approach

TL;DR

This paper develops a chaos expansion method for solving stochastic partial differential equations driven by colored noise, introducing innovative approaches to handle Wick products and providing recursive approximations with error bounds.

Contribution

It introduces a novel chaos expansion technique for SPDEs with colored noise, addressing the challenges posed by Wick products and offering practical approximation methods.

Findings

Derived a chaos expansion for the solution

Studied truncations of the chaos series

Provided recursive approximation with error bounds

Abstract

An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.