A Haar-like Construction for the Ornstein Uhlenbeck Process

TL;DR

This paper introduces a Haar-like basis for the Ornstein-Uhlenbeck process that retains key properties of the Haar basis for Brownian motion, enabling efficient finite-step evaluations at dyadic rationals.

Contribution

It develops a novel basis for the Ornstein-Uhlenbeck process with properties similar to Haar functions, facilitating finite evaluations and preserving compact support.

Findings

Basis elements approach Haar functions asymptotically

Finite expansion at dyadic rationals

Preserves compact support and nested intervals

Abstract

The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation and connections to asymptotic scale invariance.