Paper: arXiv 2401.11958

Authors: Daniel Kršek, Gudmund Pammer

Abstract

We investigate duality and existence of dual optimizers for several adapted optimal transport problems under minimal assumptions. This includes the causal and bicausal transport, the causal and bicausal barycenter problem, and a multimarginal problem incorporating causality constraints. Moreover, we characterize polar sets in the causal and bicausal setting and discuss applications of our results in robust finance. We consider a non-dominated model of several financial markets where stocks are traded dynamically, but the joint stock dynamics are unknown. We show that a no-arbitrage assumption naturally leads to sets of multicausal couplings. Consequently, computing the robust superhedging price is equivalent to solving an adapted transport problem, and finding a superhedging strategy means solving the corresponding dual.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly sophisticated theoretical work on adapted optimal transport, dual attainment, and polar set characterization with applications to model-independent superhedging. While math complexity and theoretical novelty are exceptionally high, it is a purely theoretical paper lacking empirical data, backtests, or code implementation.

Research Flowchart

  flowchart TD
    A["Research Goal: Establish duality & dual attainment for adapted optimal transport under minimal assumptions"] --> B["Input: Non-dominated market models, dynamic stock trading, & causality constraints"]
    B --> C["Core Methodology: Analyze causal/bicausal transport, barycenter problems, & characterize polar sets"]
    C --> D["Application & Computation: Map robust financial superhedging to adapted transport duality"]
    D --> E["Key Outcomes: Duality holds for adapted transport; robust superhedging price equals optimal transport cost & dual strategy exists"]

Cited References (50)

  1. Multicausal Transport: Barycenters and Dynamic Matching (2024) — link
  2. Fitted Value Iteration Methods for Bicausal Optimal Transport — → 2306.12658
  3. Collective arbitrage and the value of cooperation — → 2306.11599
  4. Sensitivity of Multiperiod Optimization Problems with Respect to the Adapted Wasserstein Distance (2023) — link
  5. An extension of martingale transport and stability in robust finance — → 2304.09551
  6. Optimal Stopping via Distribution Regression: a Higher Rank Signature Approach (2023) — link
  7. Denseness of biadapted Monge mappings (2022) — link
  8. Computational methods for adapted optimal transport (2022) — link
and 42 more
  1. The Wasserstein space of stochastic processes (2021) — link
  2. Weak transport for non‐convex costs and model‐independence in a fixed‐income market (2020) — link
  3. Adapted topologies and higher rank signatures (2020) — link
  4. Cournot-Nash Equilibrium and Optimal Transport in a Dynamic Setting (2020) — link
  5. All adapted topologies are equal (2019) — link
  6. Adapted Wasserstein distances and stability in mathematical finance (2019) — link
  7. Robust expected utility maximization with medial limits (2017) — link
  8. Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization (2016) — link
  9. Causal Transport in Discrete Time and Applications (2016) — link
  10. Multistage Stochastic Optimization (2014) — link
  11. ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014) — link
  12. Optimal transportation under controlled stochastic dynamics (2013) — link
  13. Arbitrage and duality in nondominated discrete-time models (2013) — link
  14. Robust hedging with proportional transaction costs (2013) — link
  15. A Stochastic Control Approach to No-Arbitrage Bounds Given Marginals, with an Application to Lookback Options (2013) — link
  16. Pathwise construction of stochastic integrals (2011) — link
  17. Model-independent bounds for option prices—a mass transport approach (2011) — link
  18. Barycenters in the Wasserstein Space (2011) — link
  19. Quasi-sure Stochastic Analysis through Aggregation (2010) — link
  20. Matching for teams (2010) — link
  21. A General Duality Theorem for the Monge–Kantorovich Transport Problem (2009) — link
  22. Optimal and better transport plans (2008) — link
  23. MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS (2006) — link
  24. On the Translocation of Masses (2006) — link
  25. The Mathematics of Arbitrage (2006) — link
  26. What is dynamic programming? (2004) — link
  27. Topics in Optimal Transportation (2003) — link
  28. Robust hedging of the lookback option (1998) — link
  29. The geometry of optimal transportation (1996) — link
  30. Uncertain volatility and the risk-free synthesis of derivatives (1995) — link
  31. Pricing and hedging derivative securities in markets with uncertain volatilities (1995) — link
  32. Polar Factorization and Monotone Rearrangement of Vector-Valued Functions (1991) — link
  33. Duality theorems for marginal problems (1984) — link
  34. On the optimal mapping of distributions (1984) — link
  35. Measurable Selections of Extrema (1973) — link
  36. A generalization of a problem of Steinhaus (1967) — link
  37. Appendix F: A Komlos-type lemma from probability theory (2018)
  38. A Distance For Multistage Stochastic Optimization Models (2012) — link
  39. Classical descriptive set theory (1987) — link
  40. Décomposition polaire et réarrangement monotone des champs de vecteurs (1987)
  41. Probabilities and metrics (1976)
  42. On a space of completely additive functions (1958)

3 of 50 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.