Scaling Limits of Processes with Fast Nonlinear Mean Reversion

TL;DR

This paper establishes the limiting behavior of integrals of Itô processes with rapid nonlinear mean reversion, showing they can be approximated by averaging against their invariant measure, which aids in financial modeling.

Contribution

It introduces new scaling limit results for nonlinear mean-reverting processes, extending the understanding of their asymptotic behavior under fast reversion regimes.

Findings

Processes converge uniformly in probability

Invariant measure averaging simplifies analysis

Results support portfolio optimization models

Abstract

We derive scaling limits for integral functionals of It\^o processes with fast nonlinear mean-reversion speed. We show that in these limits, the fast mean-reverting process is "averaged out" by integrating against its invariant measure. These convergence results hold uniformly in probability and, under mild integrability conditions, also in $\mathcal{S}^p$. They are a crucial building block for the analysis of portfolio choice models with small superlinear transaction costs, carried out in the companion paper of the present study.