Kullback-Leibler Barycentre of Stochastic Processes

TL;DR

This paper introduces a method to combine multiple expert diffusion process models into a single optimal model using Kullback-Leibler barycentres, with explicit formulas and deep learning algorithms for practical approximation.

Contribution

It provides the first explicit representation of the Kullback-Leibler barycentre for stochastic processes and develops two deep learning algorithms for efficient approximation of the optimal combined model.

Findings

Existence and uniqueness of the barycentre model established.

Explicit Radon--Nikodym derivative representation derived.

Two deep learning algorithms successfully approximate the optimal drift.

Abstract

We consider the problem where an agent aims to combine the views and insights of different experts' models. Specifically, each expert proposes a diffusion process over a finite time horizon. The agent then combines the experts' models by minimising the weighted Kullback--Leibler divergence to each of the experts' models. We show existence and uniqueness of the barycentre model and prove an explicit representation of the Radon--Nikodym derivative relative to the average drift model. We further allow the agent to include their own constraints, resulting in an optimal model that can be seen as a distortion of the experts' barycentre model to incorporate the agent's constraints. We propose two deep learning algorithms to approximate the optimal drift of the combined model, allowing for efficient simulations. The first algorithm aims at learning the optimal drift by matching the change of measure, whereas the second algorithm leverages the notion of elicitability to directly estimate the value function. The paper concludes with an extended application to combine implied volatility smile models that were estimated on different datasets.