arXiv subjects
Publications and source records attributed to Andrey Pilipenko.
We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.
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.
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.
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.
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.
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.