Differential Equations Driven by Gaussian Signals II

Abstract

Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a "Gaussian rough path" gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sample path regularity worse than Brownian motion. The purpose of this sequel paper is to establish convergence of Karhunen-Loeve approximations in rough path metrics. Particular care is necessary since martingale arguments are not enough to deal with third iterated integrals. An abstract support criterion for approximately continuous Wiener functionals then gives a description of the support of Gaussian rough paths as the closure of the (canonically lifted) Cameron-Martin space.