Reciprocal Specific Relative Entropy between Continuous Martingales

TL;DR

This paper introduces a new divergence measure for continuous martingales called reciprocal specific relative entropy, and identifies the neutral Wright-Fisher diffusion as the optimal model in a specific minimization problem.

Contribution

The paper defines the reciprocal specific relative entropy for continuous martingales and characterizes the optimizer as the neutral Wright-Fisher diffusion in a minimization problem.

Findings

The neutral Wright-Fisher diffusion minimizes the reciprocal specific relative entropy.

The new divergence measure offers a novel perspective on martingale comparison.

The diffusion is uniquely salient in a perturbed variance minimization problem.

Abstract

We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets Aldous (2013)). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.