Coupling and tracking of regime-switching martingales

TL;DR

This paper introduces two explicit couplings for regime-switching martingales that extend classical couplings, optimizing coupling time and tracking error, with proofs based on Bellman principle and counterexamples to conjectured optimality.

Contribution

It develops generalized mirror and synchronous couplings for regime-switching martingales, extending classical couplings and analyzing their optimality properties.

Findings

Generalized mirror coupling minimizes coupling time and maximizes tracking error.

Generalized synchronous coupling maximizes coupling time and minimizes tracking error.

Counterexamples show these couplings are not always optimal among broader classes.

Abstract

This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales. The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The proofs are based on the Bellman principle. We give counterexamples to the conjectured optimality of the two couplings amongst a wider classes of stochastic integrals.