Remarks on differentiability in the initial data for stochastic reflecting flow

TL;DR

This paper investigates the differentiability of stochastic reflecting flows in a half-plane, providing a representation of derivatives and an example where the flow is not locally continuously differentiable.

Contribution

It introduces a matrix product representation of derivatives and constructs a specific example demonstrating non-differentiability in reflecting flows.

Findings

Derivative represented as an infinite matrix product

Constructed example of non-differentiable reflecting flow

Insights into differentiability properties of reflected SDEs

Abstract

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.