Abstract
The nonlinear time dependent Brownian motion, B(g(t)), is the object of the present paper. The time-rescaling function, g(t), may be any function continuous over a finite positive time interval ranging from O to a fixed T, and such that g(0)=0. We firstly prove B(g(t)) to be equivalent to a white—noise integral having zero mean and a time—dependent variance. Secondly we find the eigenfunction (Karhunen — Loève) expansion of B(g(t)). The eigenfunctions are Bessel functions of the first kind, with a suitable time function for argument, and they are multiplied by another time function. The eigenvalues virtually are determined by the zeros of the Bessel functions.
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© 1983 D. Reidel Publishing Company
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Maccone, C. (1983). Eigenfunction Expansion for the Nonlinear Time Dependent Brownian Motion. In: Bucy, R.S., Moura, J.M.F. (eds) Nonlinear Stochastic Problems. NATO ASI Series, vol 104. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7142-4_33
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DOI: https://doi.org/10.1007/978-94-009-7142-4_33
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-7144-8
Online ISBN: 978-94-009-7142-4
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