Solving SPDEs driven by colored noise: A chaos approach

Abstract

Abstract. An Itô-Skorokhod bilinear 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 Itô-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.1. Introduction. Stochastic differential equations driven by Gaussian white noise are well-studied; see, for example, the book [35] for ordinary differential equations and the book [37] for equations with partial derivatives. The underlying stochastic process in these equations is the standard Brownian motion W , which is a square-integrable Gaussian martingale with continuous trajectories and independent increments. A lot less is known about equations driven by colored noise, when the underlying process is still Gaussian, but no longer has independent increments. An important example is the fractional Brownian motion W H , H ∈ (0, 1), which coincides with the standard Brownian motion W for H = 1/2 and is not a semi-martingale for all H = 1/2. It is the lack of the semi-martingale property that makes the analysis difficult at the very basic level, the definition of the corresponding stochastic integral. Several versions of the stochastic integral with respect to W H have been proposed [1,12,13,14,23,26,38]. Unlike the standard Brownian motion, different approaches such as Itô-type vs. Stratonovichtype integral or path-wise vs. mean-square definition, become much more difficult to reconcile. The paper by V. Pipiras and M. Taqqu [36] describes the main difference