Scaling limit of d-inverse of Brownian motion with functional drift

TL;DR

This paper investigates the scaling limits of the d-inverse of Brownian motion with functional drift, revealing that, aside from degenerate cases, limits include d-inverses of Brownian motion without drift, with explosion, or with power drift.

Contribution

It characterizes the possible scaling limits of the d-inverse of Brownian motion with functional drift, expanding understanding of its asymptotic behavior.

Findings

Limits include d-inverses of Brownian motion without drift

Existence of limits with explosion in finite time

Limits with power drift identified

Abstract

The d-inverse is a generalized notion of inverse of a stochastic process having a certain tendency of increasing expectations. Scaling limit of the d-inverse of Brownian motion with functional drift is studied. Except for degenerate case, the class of possible scaling limits is proved to consist of the d-inverses of Brownian motion without drift, one with explosion in finite time, and one with power drift.