A Haar-like Construction for the Ornstein Uhlenbeck Process |
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Authors: | Thibaud Taillefumier Marcelo O Magnasco |
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Institution: | (1) Laboratory of Mathematical Physics, The Rockefeller University, 10021 New York, NY, USA |
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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.
Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users. |
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Keywords: | Ornstein-Uhlenbeck process Brownian motion Haar basis |
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