Izyumtseva O. L.
Ukr. Mat. Zh. - 2018. - 70, № 12. - pp. 1587-1614
We prove the existence of a multiple local time of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernel, describe its Ito – Wiener expansion and establish the Clark representation.
Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 290-340
This survey article is devoted to the local times of self-intersection as the most important geometric characteristics of random processes. The trajectories of random processes are, as a rule, very nonsmooth curves. This is why to characterize the geometric shape of the trajectory it is impossible to use the methods of differential geometry. Instead of this, one can consider the local times of self-intersection showing how much time the process stays in “small” vicinities of its self-crossing points. In our paper, we try to describe the contemporary state of the theory of local times of self-intersection for Gaussian and related processes. Different approaches to the definition, investigation, and application of the local times of self-intersection are considered.
Renormalization constant for the local times of self-intersections of a diffusion process in the plane
Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1489–1498
We study the local times of self-intersection of a diffusion process in the plane. Our main result is connected with the investigation of the asymptotic behavior of the renormalization constant of this local time.