Stochastic flows with reflection

TL;DR

This paper investigates the topological properties and Sobolev differentiability of stochastic flows with reflection in the positive orthant, and studies the absolute continuity of associated measure-valued processes.

Contribution

It provides new results on the differentiability and measure properties of stochastic flows with boundary reflection in Euclidean space.

Findings

Sobolev differentiability in initial conditions established

Absolute continuity of measure-valued processes shown

Topological properties of reflected stochastic flows analyzed

Abstract

Some topological properties of stochastic flow $\varphi_t(x)$ 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 $\mu\circ\varphi_t^{-1}$, where $\mu\ll\lambda^d,$ is studied.