The Lamperti Transforms of Self-Similar Gaussian Processes and Their Exponentials |
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Authors: | Muneya Matsui Narn-Rueih Shieh |
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Affiliation: | 1. Department of Business Administration , Nanzan University , Nagoya , Japan;2. Department of Mathematics , National Taiwan University , Taipei , Taiwan |
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Abstract: | We present results on the second order behavior and the expected maximal increments of Lamperti transforms of self-similar Gaussian processes and their exponentials. The Ornstein Uhlenbeck processes driven by fractional Brownian motion (fBM) and its exponentials have been recently studied in Ref.[ 20 Matsui , M. ; Shieh , N.-R. On the exponentials of fractional Ornstein-Uhlenbeck processes . Electron. J. Probab. 2009 , 14 , 594 – 611 .[Crossref], [Web of Science ®] , [Google Scholar] ] and Ref.[ 21 Matsui , M. ; Shieh , N.-R. On the exponential process associated with a CARMA-type process. Stochastics , 2012 . doi: 10.1080/17442508.2012.654791 .[Taylor &; Francis Online] , [Google Scholar] ], where we essentially make use of some particular properties, e.g., stationary increments of fBM. Here, the treated processes are fBM, bi-fBM, and sub-fBM; the latter two are not of stationary increments. We utilize decompositions of self-similar Gaussian processes and effectively evaluate the maxima and correlations of each decomposed process. We also present discussion on the usage of the exponential stationary processes for stochastic modeling. |
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Keywords: | Bi-fractional Brownian motion Exponential Processes Fractional Brownian motion Lamperti transforms Self-similarity Sub-fractional Brownian motion |
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