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Solutions of linear and semilinear distributed parameter equations with a fractional Brownian motion
Author(s) -
Duncan T. E.,
Maslowski B.,
PasikDuncan B.
Publication year - 2009
Publication title -
international journal of adaptive control and signal processing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.73
H-Index - 66
eISSN - 1099-1115
pISSN - 0890-6327
DOI - 10.1002/acs.1050
Subject(s) - fractional brownian motion , mathematics , brownian motion , hurst exponent , mathematical analysis , estimator , hilbert space , linear equation , statistics
In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parameter H ∈ (0, 1). For H = ½ the process is Brownian motion. Solutions of these linear and semilinear equations are given for each H ∈ (0, 1) with the assumptions differing for the cases H ∈ (0, ½) and H ∈ (½, 1). For the linear equations, the solutions are mild solutions and limiting Gaussian measures are characterized. For the semilinear equations, the solutions are either mild or weak. The weak solutions are obtained by transforming the measure of the associated linear equation by a Radon–Nikodym derivative (likelihood function). An application to identification is given by obtaining a strongly consistent family of estimators for an unknown parameter in a linear equation with distributed noise or boundary noise. Copyright © 2008 John Wiley & Sons, Ltd.