A Haar-like Construction for the Ornstein Uhlenbeck Process

Details A Haar-like Construction for the Ornstein Uhlenbeck Process A Haar-like Construction for the Ornstein Uhlenbeck Process

2007-09-11

arxiv.org/abs/0709.1726v1

Mathematics

Probability

20 pages, 3 figures

Scientific paper

10.1007/s10955-008-9545-8

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.

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