Summary
In this paper, we study oscillatory stochastic integrals of the form\(\Gamma (\lambda ) = \int\limits_0^\infty {exp(i \lambda B_s } )g(s)d s\) where λ is a non zero parameter andg a square integrable function. We study integrability properties of Γ(λ) and its behavior as a function of λ, using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables Γ(λ) based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.
We consider the random measure, image by the Brownian motion of the unbounded measure 1[0,∞] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.
Finally, under a stronger integrability condition ong, we show the existence of a density for the law of Γ(λ), using Malliavin's calculus.
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Donati-Martin, C. Transformation de Fourier et temps d'occupation browniens. Probab. Th. Rel. Fields 88, 137–166 (1991). https://doi.org/10.1007/BF01212557
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DOI: https://doi.org/10.1007/BF01212557