On the specific relative entropy between martingale diffusions on the line

TL;DR

This paper investigates the specific relative entropy between martingale diffusions on the real line, providing partial validation of a conjecture relating it to quadratic variations, with implications for large deviations and transport inequalities.

Contribution

It offers a partial proof of a conjecture linking specific relative entropy of martingale laws to their quadratic variations for well-behaved diffusions.

Findings

Validated the conjecture for certain martingale diffusions on the line.

Connected specific relative entropy to quadratic variations in this context.

Enhanced understanding of entropy in stochastic process laws.

Abstract

The specific relative entropy, introduced by N. Gantert, allows to quantify the discrepancy between the laws of potentially mutually singular measures. It appears naturally as the large deviations rate function in a randomized version of Donsker's invariance principle, as well as in a novel transport-information inequality recently derived by H. Foellmer. A conjecture, put forward by the aforementioned authors, concerns a closed form expression for the specific relative entropy between continuous martingale laws in terms of their quadratic variations. We provide a first partial result in this direction, by establishing this conjecture in the case of well-behaved martingale diffusions on the line.