Approximating Multiple Itô Integrals with “Wide-Band Width” Processes, and Application to Filtering Theory

Abstract

Let nε(·) denote a "wide-band widt h" vector valued process. We are concerned with limits of ∫0t∫0τnε'(τ)L(τ,s)nε(s)dτds, and for the m-multiple integral case. A weak convergence result is obtained, and the “correction” terms exhibited. An application to a likelihood functional and hypothesis testing problem is given. There, the weak convergence result (rather than mere convergence of finite dimensional distributions) is essential if the limit approximation is to make sense as an approximation to the likelihood functional. The correction terms depend only on the limit (as ε → 0) of the correlation function of the (renormalized) nε(·).