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Andrey Pilipenko

Publications and source records attributed to Andrey Pilipenko.

44 records · Page 3Linked to original sources
arXiv2015

On a limit behavior of a sequence of Markov processes perturbed in a neighborhood of a singular point

We study a limit behavior of a sequence of Markov processes (or Markov chains) such that their distributions outside of any neighborhood of a "singular" point attract to some probability law. In any neighborhood of this point the behavior may be irregular. As an example of the general result we consider a symmetric random walk with the unit jump that is perturbed in a neighborhood of 0. The invariance principle is obtained under standard scaling of time and space. The limit process turns out to be a skew Brownian motion.

math.PR↗
arXiv2015

A functional limit theorem for locally perturbed random walks

A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. Assuming that the particle cannot jump over the membrane we prove that the weak scaling limit of the particle position is a skew Brownian motion with parameter $γ\in [-1,1]$. The path of a skew Brownian motion is obtained by taking each excursion of a reflected Brownian motion, independently of the others, positive with probability $2^{-1}(1+γ)$ and negative with probability $2^{-1}(1-γ)$. To prove the weak convergence result we offer a new approach which is based on the martingale characterization of a skew Brownian motion. Among others, this enables us to provide the explicit formula for the parameter $γ$. In the previous articles the explicit formulae for the parameter have only been obtained under the assumption that outside the membrane the particle performs unit jumps.

math.PR↗
arXiv2014

A remark on the paper "Renorming divergent perpetuities"

Let $(ξ_k)$ and $(η_k)$ be infinite independent samples from different distributions. We prove a functional limit theorem for the maximum of a perturbed random walk $\underset{0\leq k\leq n}{\max}\,(ξ_1+\ldots+ξ_k+η_{k+1})$ in a situation where its asymptotics is affected by both $\underset{0\leq k\leq n}{\max}\,(ξ_1+\ldots+ξ_k)$ and $\underset{1\leq k\leq n}{\max}\,η_k$ to a comparable extent. This solves an open problem that we learned from the paper "Renorming divergent perpetuities" by P. Hitczenko and J. Wesołowski.

math.PR↗
arXiv2013

On existence and properties of strong solutions of one-dimensional stochastic equations with an additive noise

One-dimensional stochastic differential equations with additive Lévy noise are considered. Conditions for existence and uniqueness of a strong solution are obtained. In particular, if the noise is a Lévy symmetric stable process with $α\in(1;2)$, then the measurability and boundedness of a drift term is sufficient for the existence of a strong solution. We also study continuous dependence of the strong solution on the initial value and the drift.

math.PR↗
arXiv2012

Remarks on differentiability in the initial data for stochastic reflecting flow

Stochastic flows generated by reflected SDEs in a half-plane with an additive diffusion term are considered. A derivative in the initial data is represented a.s. as an infinite product of matrices. We use this representation and construct an example of a reflecting flow with a linear drift such that it is not locally continuously differentiable.

math.PR↗
arXiv2009

On simultaneous hitting of membranes by two skew Brownian motions

We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.

math.PR↗
arXiv2008

Stochastic flows with reflection

Some topological properties of stochastic flow $φ_t(x)$ generated by stochastic differential equation in a ${\mathbb R}^d_+$ with normal reflection at the boundary are investigated. Sobolev differentiability in initial condition is received. The absolute continuity of the measure-valued process $μ\circφ_t^{-1}$, where $μ\llλ^d,$ is studied.

math.PR↗