Sufficiency on the Stock Market

TL;DR

This paper explores the connections between finance and information theory, showing that portfolio theory relates to Bregman divergences and identifying conditions where these divergences align with information divergence.

Contribution

It establishes the conditions under which portfolio theory leads to Bregman divergences and relates these to information divergence in finance.

Findings

Portfolio theory always leads to Bregman divergences.

Bregman divergence is proportional to information divergence only under specific conditions.

The results relate to the type of gambling studied by Kelly.

Abstract

It is well-known that there are a number of relations between theoretical finance theory and information theory. Some of these relations are exact and some are approximate. In this paper we will explore some of these relations and determine under which conditions the relations are exact. It turns out that portfolio theory always leads to Bregman divergences. The Bregman divergence is only proportional to information divergence in situations that are essentially equal to the type of gambling studied by Kelly. This can be related an abstract sufficiency condition.