On self-intersection local times for generalized Brownian bridges and the distance between step functions

2015/05/25 by A. A. Dorogovtsev, Dorogovtsev, A. A., Olga Izyumtseva +2
Economics, Econometrics and Finance · Mathematics · #46C05 #60G15 #60H40 #60J55 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:46C05 #msc:60G15 #msc:60H40 #msc:60J55

paper · pdf · doi:10.48550/arxiv.1505.06620

12 pages

arxiv created 2015/05/25 · openalex publication_date 2015/05/25 · arxiv updated 2015/05/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In the paper k-multiple self-intersection local time for planar Gaussian integrators generated by linear operator with nontrivial kernel is studied. In this case additional singularities arise in its formal Fourier--Wiener transform. In case k=2 the set of singularities is the finite number of points. In case k>2 it contains intervals and hyperplanes. In the first and the second cases using two different approaches related on structure of set of singularities we show that "new" singularities do not imply on the convergence of integral corresponding to the formal Fourier--Wiener transform and regularization consist of compensation of impact of diagonals as for the Wiener process.

Related

• On regularization of formal Fourier--Wiener transform of the self-intersection local time of planar Gaussian process
• Clark representation for self-intersection local times of Gaussian integrators
• Refinements of asymptotics at zero of Brownian self-intersection local times
• Hilbert-valued self-intersection local times for planar Brownian motion
• Local times of self-intersection and sample path properties of Volterra Gaussian processes