# Karhunen-Loeve Expansions for Weighted Wiener Processes and Brownian Bridges via Bessel Functions

@inproceedings{Deheuvels2003KarhunenLoeveEF,
title={Karhunen-Loeve Expansions for Weighted Wiener Processes and Brownian Bridges via Bessel Functions},
author={Paul Deheuvels and G. V. Martynov},
year={2003}
}
We provide Karhunen—Loeve expansions on (0, 1) for the processes t θ B (tρ)and t θ W (tθ)where B (•) a Brownian bridge, W(•) is a Wiener process, and ρ and θ are arbitrary real numbers such that θ > −(ρ + 1)/2. The eigenfunctions of these expansions have simple expressions in terms of Bessel functions J v(•) and Jv-1(•) of the first kind with indexes ν = ρ/(2θ + ρ + 1) and ν − 1 = −(2θ + 1)/(2θ + ρ + 1). The corresponding eigenvalues have simple expressions in terms of the positive zeros z ν,1… CONTINUE READING

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