Andrey Pilipenko

On a selection problem for small noise perturbation in the multidimensional case

The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.

On Brownian motion on the plane with membranes on rays with a common endpoint

Abstract We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one.

A functional limit theorem for excited random walks

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method of investigations based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

A limit theorem for singular stochastic differential equations

We study the weak limits of solutions to SDEs \[dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),\] where the sequence

On a Brownian motion with a hard membrane

\{a_n\}

On differentiability with respect to the initial data of the solution to an SDE with a Lévy noise and discontinuous coefficients

converges in some sense to

Optimal stopping of random sequences and processes

(c_- 1\mkern-4.5mu\mathrm{l}_{x 0})/x+\gamma\delta_0

Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures

. Here

Renewal theory. Queueing theory

\delta_0

Basic functionals of the risk theory

is the Dirac delta function concentrated at zero. A limit of