On the number of crossings of some levels by a sequence of diffusion processes


  • M.M. Osypchuk Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0001-6100-1654


diffusion process, stochastic differential equation
Published online: 2009-12-30


The limit behavior of the number of crossings of some sequence of levels by the following sequence of random variables $\xi_n(0)$, $\xi_n\left(\frac{1}{m}\right)$, $\dots$, $\xi_n\left(\frac{N}{m}\right)$, as the integers $n$, $m$, $N$ are increasing to infinity in some consistent way, is investigated, where $(\xi_n(t))_{t\ge0}$ for $n=1,2,\dots$ is a diffusion process on a real line $\mathbb{R}$ with its local characteristics (that is, drift and diffusion coefficients) $(a_n(x))_{x\in\mathbb{R}}$ and $(b_n(x))_{x\in\mathbb{R}}$ given by $a_n(x)=na(nx)$, $b_n(x)=b(nx)$ for $x\in\mathbb{R}$ and $n=1,2,\dots$ with some fixed functions $(a(x))_{x\in\mathbb{R}}$ and $(b(x))_{x\in\mathbb{R}}$.

How to Cite
Osypchuk, M. On the Number of Crossings of Some Levels by a Sequence of Diffusion Processes. Carpathian Math. Publ. 2009, 1, 191-196.