Enhanced Gaussian processes and applications

We propose some construction of enhanced Gaussian processes using Karhunen-Loeve expansion. We obtain a characterization and some criterion of existence and uniqueness. Using roughpath theory, we derive some Wong-Zakai Theorem. Mathematics Subject Classification. 60G15, 60G17. Received July 2, 2007. Revised October 5, 2007. 1. Generalities In [13] Lyons developed a general theory of differential equations of the form dyt = f(yt)dxt. (1.1) Classical integration/ODE theory gives a meaning to such differential equations when x has bounded variation. Lyons extended this notion to the case when x is a path with values in a Banach space B, and of finite p-variation, p ≥ 1. To do so, one needs first to lift x to a path of finite p-variation in the free nilpotent group of B. In other words, one needs to define and make a choice for the “iterated integrals” of order less than or equal to [p] of x. We refer the reader to, for example, [11,13,14]. In this paper, our aim is to work towards the study of a “natural” p-rough path process lying above an arbitrary Gaussian process. We simplify the problem by only looking at lift in the free nilpotent group of step 2, i.e. we are just looking at the Lévy area of Gaussian processes. This was already done by Lévy in 1950 for Brownian motion, see [12] or more recently [10] and [6], and for fractional Brownian motion, see [5]or [15]. Moreover, Biane and Yor, in [1] have constructed the Lévy area using the expansion of Brownian motion in the basis of Legendre polynoms. Karhunen-Loeve expansion Theorem provide a natural way to approximate paths of a Gaussian process by a smooth process. This paper is devoted to study how its expansion allow to lift R-valued Gaussian process x to a path x with values in some free nilpotent of step 2 group over R (or in other words, how to construct the Lévy area of x, i.e. the second iterated integral of x). We also show that if the process x with some area process satisfies some quite natural conditions, then x will be the limit of the lift of the Karhunen-Loeve approximations of x. The proof of the convergence of Karhunen-Loeve expansion Theorem or of some properties on Gaussian processes relies on the convex property of the vector spaces. The free nilpotent group of step 2 do not share this property. In the first part of this paper, we give a proof of a weak version the Karhunen-Loeve expansion