Paper: arXiv 2407.04860

Authors: Sebastian Jaimungal, Silvana M. Pesenti

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework for finding Kullback-Leibler barycentres of continuous-time diffusion processes with constraints. It combines deep stochastic analysis with modern machine learning techniques (deep learning for optimal drift approximation) and demonstrates an application to volatility smile modeling.

Research Flowchart

  flowchart TD
    A[Research Goal: Combine Expert Diffusion Models via KL Barycentre] --> B[Inputs: Experts' Diffusion Models & Agent Constraints]
    B --> C[Theoretical Derivation: Prove Existence, Uniqueness & Radon-Nikodym Derivative]
    C --> D[Deep Learning Algorithms]
    D --> D1[Alg 1: Match Change of Measure]
    D --> D2[Alg 2: Value Function via Elicitability]
    D1 & D2 --> E[Application: Aggregate Implied Volatility Smile Models]
    E --> F[Outcome: Distortion-Adjusted Optimal Combined Drift Model]

Cited References (38)

  1. Phase portrait of the matrix Riccati equation (1986)
  2. Non—blow—up conditions for riccati—type matrix differential and difference equations (2000)
  3. On an identity for stochastic integrals (1973)
  4. Bicausal optimal transport for SDEs with irregular coefficients (2024) — link
  5. Adapted optimal transport between G aussian processes in discrete time (2025)
  6. Reinforcement learning with dynamic convex risk measures (2024)
  7. Conditionally elicitable dynamic risk measures for deep reinforcement learning (2023)
  8. ScoreFusion: fusing score-based generative models via K ullback- L eibler barycenters (2024) — link
and 30 more
  1. Carbon Finance: A Risk Management View (2021)
  2. Using dynamic time warping to find patterns in time series (1994)
  3. Prediction games and arcing algorithms (1999)
  4. FuNVol: multi-asset implied volatility market simulator using functional principal components and neural SDE s (2024)
  5. Bayesian learning with W asserstein barycenters (2022)
  6. Wasserstein barycenter model ensembling (2019) — link
  7. A Geometric Approach To Asset Allocation With Investor Views — → 2406.01199
  8. Dynamic portfolio management with views at multiple horizons (2016)
  9. Entropy Pooling with Discrete Weights in a Time-Dependent Setting (2019)
  10. Minimal K ullback– L eibler Divergence for Constrained L 'e vy– I t ^o Processes (2024)
  11. Making and evaluating point forecasts (2011)
  12. Wasserstein barycenters are NP -hard to compute (2022)
  13. Targeted transfer learning through distribution barycenter medium for intelligent fault diagnosis of machines with data decentralization (2024)
  14. Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation (2005)
  15. Bagging predictors (1996)
  16. Prediction games and arcing algorithms (1999)
  17. Neural network ensembles (1990)
  18. Neural network ensemble strategies for financial decision applications (2005)
  19. Forecasting crude oil price with an EMD -based neural network ensemble learning paradigm (2008)
  20. Classification of genomic signals using dynamic time warping (2013)
  21. On the hidden M arkov model and dynamic time warping for speech recognition—A unified view (1984)
  22. Trimmed fuzzy clustering of financial time series based on dynamic time warping (2021)
  23. I-divergence geometry of probability distributions and minimization problems (1975)
  24. Stressing dynamic loss models (2024)
  25. Adapted W asserstein distance between the laws of SDE s (2022) — link
  26. On a W asserstein-type distance between solutions to stochastic differential equations (2019)
  27. Risk Budgeting Allocation for Dynamic Risk Measures — → 2305.11319
  28. Sensitivity measures based on scoring functions (2023)
  29. Multicausal transport: barycenters and dynamic matching (2024) — link
  30. The W asserstein space of stochastic processes (2024)

2 of 38 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.