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Local time and related sample paths of filtered white noises. (English) Zbl 1144.60029

The author studies the properties of the local times of the Filtered White Noise (FWN), which is a Gaussian process with representation

X(t)= a(t,λ)(e itλ -1) |λ| 1/2+H dW(λ),t[0,1],

where 0<H<1. The main result of the paper states that FWN has (a.s) jointly continuous local time L(t,x) which is Hölder continuous, with certain Hölder exponents. The approach used in the proof is based on the Berman concept of local nondeterminism, see [S. M. Berman, Ann. Math. Stat. 41, 1260–1272 (1970; Zbl 0204.50501]. Further the author proves the Chung’s law of iterated logarithm for FWN, and uses this result to find the lower bound for the moduli of continuity of the local time. See also the related papers [B. Boufoussi, M. Dozzi, R. Guerbaz, Bernoulli 13, No. 3, 849–867 (2007; Zbl 1138.60032) and Stochastics 78, No. 1, 33–49 (2006; Zbl 1124.60061)].

60G15Gaussian processes
60G17Sample path properties