About the infinite dimensional skew and obliquely reflected Ornstein-Uhlenbeck process

TL;DR

This paper constructs and analyzes an infinite-dimensional skew and obliquely reflected Ornstein-Uhlenbeck process in a Hilbert space, extending classical finite-dimensional reflection concepts with new existence, uniqueness, and example results.

Contribution

It introduces a novel infinite-dimensional framework for oblique and skew reflections of Ornstein-Uhlenbeck processes, including a Skorokhod decomposition and concrete examples.

Findings

Established existence and uniqueness of solutions under certain conditions.

Developed a Skorokhod type decomposition for the process.

Provided examples including the infinite-dimensional p-skew reflected process.

Abstract

Based on an integration by parts formula for closed and convex subsets $\Gamma$ of a separable real Hilbert space $H$ with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein-Uhlenbeck process (perturbed by a bounded drift $B$) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary $\partial \Gamma$ is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vector) at the reflection point. In case of normal reflection at the boundary of a regular convex set and under some monotonicity condition on $B$, we prove the existence and uniqueness of a strong solution to the corresponding SDE. Subsequently, we consider an increasing sequence $(\Gamma_{\alpha_k})_{k\in\mathbb{Z}}$ of closed and convex subsets of $H$ and the skew reflection problem at the boundaries of this sequence. We present concrete examples and obtain as a special case the infinite dimensional analogue of the $p$-skew reflected Ornstein-Uhlenbeck process.