Kullback-Leibler Barycentre of Stochastic Processes

TL;DR

This paper considers the problem where an agent aims to combine the views and insights of different experts' models, and proposes two deep learning algorithms to approximate the optimal drift of the combined model, allowing for efficient simulations.

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.

Figures (7)

• Figure 1: A representation of problem \ref{['opt:P']}, where ${\mathcal{C}}$ denotes the set of measures that attain the problem constraints.
• Figure 1: Illustration that first finding the barycenter model and imposing constraints is equivalent to directly imposing the constraints and minimizing the weighted KL divergence.
• Figure 1: Top: (left) comparison of loss as a function of iteration, and (right) scatter plot of obtain RN derivative versus target, both for various time discretisations. Bottom: evolution under the expert models, average model, and optimal model.
• Figure 2: (top) sample paths under the expert models, the average drift model, and the optimal measure. (bottom) The left and middle panels show the evolution of the constraints as the two learning algorithms proceed. The right panel shows the histogram of the difference between the RN derivative of the learnt models minus the target RN derivative $\frac{{\rm d}{\mathbb{Q}}[\theta_{\eta^*}]}{{\rm d}{\mathbb{Q}}[{\bar{\mu}}]}$.
• Figure 3: Time series of the (normalised) functional basis coefficients from implied volatility smiles.

Theorems & Definitions (25)

• Definition 2.1
• Definition 3.1: Barycentre measure
• Theorem 3.2: Barycentre Drift and Value Function
• Proof 1
• Proposition 3.3: Pure barycentre Measure Change
• Proof 2
• Proposition 3.4: Asymptotic Expansion.
• Proof 3
• Corollary 3.5: Perturbed Drift.
• Proof 4