On Sdes For Bessel Processes In Low Dimension And Path-dependent Extensions

TL;DR

This paper characterizes low-dimensional Bessel processes as solutions to SDEs with distributional drift and introduces path-dependent extensions, expanding the understanding of their stochastic properties.

Contribution

It provides a novel characterization of low-dimensional Bessel processes via SDEs with distributional drift and defines path-dependent Bessel processes through path-dependent SDEs.

Findings

Unique solutions to SDEs with distributional drift for low-dimensional Bessel processes

Introduction of path-dependent Bessel processes and their characterization

Extension of Bessel process theory to non-semimartingale cases

Abstract

The Bessel process in low dimension (0 $\le$ $\delta$ $\le$ 1) is not an It{\^o} process and it is a semimartingale only in the cases $\delta$ = 1 and $\delta$ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift.