Paper: arXiv 2601.19511

Authors: Johannes Langner, Gregor Svindland

Abstract

This paper assumes a robust stochastic model where a set $\mathcal{P}$ of probability measures replaces the single probability measure of dominated models. We introduce and study $\mathcal{P}$-sensitive functions defined on robust function spaces of random variables. We show that $\mathcal{P}$-sensitive functions are precisely those that admit a representation via so-called functional localization. The theory is applied to solving robust optimization problems, to convex risk measures, and to the study of no arbitrage in robust one-period financial models.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous functional-analytic framework for handling Knightian uncertainty using non-dominated sets of probability measures and P-sensitive functions. While mathematically sophisticated and complete in its theoretical derivations, it is a purely theoretical paper with no empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Robust Stochastic Models<br>via P-Sensitive Functions] --> B[Input: Set of Probability Measures P &<br>Robust Function Spaces]
    B --> C[Define & Characterize<br>P-Sensitive Functions]
    C --> D[Develop Functional Localization<br>Representation Theorem]
    D --> E[Outcomes & Applications:<br>1. Solved Robust Optimization Problems<br>2. Evaluated Convex Risk Measures<br>3. Characterized No-Arbitrage in 1-Period Models]

Cited References (137)

  1. A Model-Free Version of the Fundamental Theorem of Asset Pricing and the Super - Replication Theorem (2013) — link
  2. Infinite Dimensional Analysis: A Hitchhiker’s Guide (2006) — link
  3. Locally Solid Riesz Spaces with Applications to Economics (2003) — link
  4. Positive Operators (2006) — link
  5. Contracting on Ambiguous Prospects (2017) — link
  6. A pointwise bipolar theorem (2019) — link
  7. Pathwise superhedging on prediction sets (2020) — link
  8. Duality theory for robust utility maximisation (2021) — link
and 129 more
  1. On Arbitrage and Duality under Model Uncertainty and Portfolio Constraints (2015) — link
  2. Duality and General Equilibrium Theory Under Knightian Uncertainty (2018) — link
  3. Risk measuring under model uncertainty (2012) — link
  4. No-arbitrage with multiple-priors in discrete time (2020) — link
  5. Arbitrage and duality in nondominated discrete-time models (2015) — link
  6. A bipolar theorem for $ L ^0_+( \Omega , \mathcal F , \mathbb P )$ (1999)
  7. Pointwise Arbitrage Pricing Theory in Discrete Time (2019) — link
  8. Model-free superhedging duality (2017) — link
  9. Arbitrage-free modeling under Knightian uncertainty (2020) — link
  10. Viability and Arbitrage Under Knightian Uncertainty (2021) — link
  11. On Robust Fundamental Theorems of Asset Pricing in Discrete Time (2024) — link
  12. Super‐replication with transaction costs under model uncertainty for continuous processes (2022) — link
  13. Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces (2012) — link
  14. The Mathematics of Arbitrage (2006) — link
  15. Function Spaces and Capacity Related to a Sublinear Expectation: Application to G - B rownian Motion Paths (2011) — link
  16. Real Analysis and Probability (2002) — link
  17. Convex Analysis and Variational Problems (1999) — link
  18. Stochastic Finance (2016) — link
  19. Surplus-Invariant Risk Measures (2020) — link
  20. On the Existence of a Minimal Sufficient Subfield (1974) — link
  21. Robust pricing–hedging dualities in continuous time (2018) — link
  22. On pathwise stochastic integration (1995) — link
  23. The asymptotic elasticity of utility functions and optimal investment in incomplete markets (1999) — link
  24. Necessary and sufficient conditions in the problem of optimal investment in incomplete markets (2003)
  25. Dual representation of monotone convex functions on $ L ^0$ (2011) — link
  26. Bipolar theorems for sets of nonnegative random variables (2025) — link
  27. Model Uncertainty: A Reverse Approach (2022) — link
  28. Separability Versus Robustness of Orlicz Spaces: Financial and Economic Perspectives (2022) — link
  29. Markov-Nash equilibria in mean-field games under model uncertainty (2024) — link
  30. Fatou closedness under model uncertainty (2018) — link
  31. Financial options and statistical prediction intervals (2003) — link
  32. Measurability of semimartingale characteristics with respect to the probability law (2014) — link
  33. Superreplication under model uncertainty in discrete time (2014) — link
  34. A unified framework for robust modelling of financial markets in discrete time (2021) — link
  35. Financial economics without probabilistic prior assumptions (2015) — link
  36. Quasi-sure Stochastic Analysis through Aggregation (2011) — link
  37. Comparison of Statistical Experiments (1991) — link
  38. Probability Theory: A Comprehensive Course (2014) — link
  39. Convergence of Probability Measures (1999) — link
  40. Measure Theory (2007) — link
  41. Markov decision processes under model uncertainty (2023) — link
  42. Markov–Nash Equilibria in Mean-Field Games with Discounted Cost (2018) — link
  43. Convergence of Lebesgue integrals with varying measures (1982)
  44. Learning in Mean Field Games: A Survey (2024) — link
  45. On the Convergence of Model Free Learning in Mean Field Games (2020) — link
  46. Large population stochastic dynamic games: Closed loop McKean-Vlasov sysyems and the Nash certainity equivalence principle (2006)
  47. Mean field games (2007) — link
  48. Discrete time mean-field stochastic linear-quadratic optimal control problems (2013) — link
  49. Linear-quadratic- Gaussian mean field games under high rate quantization (2013) — link
  50. Approximate Nash Equilibria in Partially Observed Stochastic Games with Mean-Field Interactions (2019) — link
  51. Robust mean field games for coupled Markov jump linear systems (2016) — link
  52. Linear Quadratic Risk-Sensitive and Robust Mean Field Games (2017) — link
  53. Risk-Sensitive Mean-Field Games (2014) — link
  54. Robust Mean Field Games (2016) — link
  55. Mean field LQG games with model uncertainty (2013) — link
  56. Discrete-time decentralized control using the risk-sensitive performance criterion in the large population regime: A mean field approach (2015) — link
  57. Nonconcave Stochastic Optimal Control in Finite Discrete Time Under Model Uncertainty (2025) — link
  58. Maxmin expected utility with non-unique prior (1989) — link
  59. Mean Field Games and Mean Field Type Control Theory (2013) — link
  60. Probabilistic Theory of Mean Field Games with Applications I-II (2018) — link
  61. Discrete time, finite state space mean field games (2010) — link
  62. Applications of Mean Field Games in Financial Engineering and Economic Theory (2020) — link
  63. Mean Field Games and systemic risk (2015) — link
  64. Mean Field Games of Timing and Models for Bank Runs (2017) — link
  65. On a mean field game approach modeling congestion and aversion in pedestrian crowds (2011) — link
  66. Dynamic driving and routing games for autonomous vehicles on networks: A mean field game approach (2021) — link
  67. Optimal Incentives to Mitigate Epidemics: A Stackelberg Mean Field Game Approach (2022) — link
  68. Contact rate epidemic control of COVID-19: an equilibrium view (2020) — link
  69. Ambiguity, Risk, and Asset Returns in Continuous Time (2002) — link
  70. Intertemporal Asset Pricing under Knightian Uncertainty (1994) — link
  71. Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio (1992) — link
  72. Notes on mean field games (from P.-L. Lions’ lectures at Coll` e ge de France ) (2011)
  73. Mean Field Games Models—A Brief Survey (2014) — link
  74. Large-Population LQG Games Involving a Major Player: The Nash Certainty Equivalence Principle (2010) — link
  75. A general characterization of the mean field limit for stochastic differential games (2016) — link
  76. A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes (2022) — link
  77. Mean field and n‐agent games for optimal investment under relative performance criteria (2019) — link
  78. From the master equation to mean field game limit theory: Large deviations and concentration of measure (2020) — link
  79. Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls (2022) — link
  80. Continuous Time Finite State Mean Field Games (2013) — link
  81. Anonymous sequential games (1988) — link
  82. Model-free mean-field reinforcement learning: Mean-field MDP and mean-field Q-learning (2023) — link
  83. A mean field approach for optimization in discrete time (2011) — link
  84. Mean Field for Markov Decision Processes: From Discrete to Continuous Optimization (2012) — link
  85. Dynamic Programming Principles for Mean-Field Controls with Learning (2023) — link
  86. Mean-Field Controls with Q -Learning for Cooperative MARL : Convergence and Complexity Analysis (2021) — link
  87. Mean-field Markov decision processes with common noise and open-loop controls (2022) — link
  88. Mean Field Markov Decision Processes (2023) — link
  89. Equilibria of dynamic games with many players: Existence, approximation, and market structure (2015) — link
  90. Mean Field Games with Ergodic cost for Discrete Time Markov Processes (2015) — link
  91. Discrete-time stochastic Stackelberg dynamic games with a large number of followers (2016) — link
  92. On general minimax theorems (1958) — link
  93. Zero-Sum Stochastic Stackelberg Games (2022) — link
  94. Computing Stackelberg Equilibria in Discounted Stochastic Games (2021) — link
  95. Convex-Concave Min-Max Stackelberg Games (2021) — link
  96. Correlated-Q learning (2003)
  97. Markov games as a framework for multi-agent reinforcement learning (1994) — link
  98. Friend-or-Foe Q-learning in General-Sum Games (2001)
  99. $Q$-Learning in a Stochastic Stackelberg Game between an Uninformed Leader and a Naive Follower (2019) — link
  100. Nash Q -learning for general-sum stochastic games (2003)
  101. Fictitious Play in Zero-Sum Stochastic Games (2022) — link
  102. Q -Learning for Distributionally Robust Markov Decision Processes (2021)
  103. Distributionally robust optimization for sequential decision-making (2019) — link
  104. Robust optimal control using conditional risk mappings in infinite horizon (2018) — link
  105. Distributionally Robust Markov Decision Processes (2012) — link
  106. Computing the optimal distributionally-robust strategy to commit to (2022) — link
  107. Decentralized robust V -learning for solving Markov games with model uncertainty (2023)
  108. Roping in Uncertainty: R obustness and Regularization in M arkov Games (2024) — link
  109. Robust Stackelberg Games via Static Output Feedback Strategy for Uncertain Stochastic Systems with State Delay (2020) — link
  110. Robust Multi-Agent Reinforcement Learning with Model Uncertainty (2020) — link
  111. Robust Q -learning algorithm for Markov decision processes under Wasserstein uncertainty (2024) — link
  112. Computational aspects of robust optimized certainty equivalents and option pricing (2020) — link
  113. Human-level control through deep reinforcement learning (2015) — link
  114. Machine Learning in Finance: From Theory to Practice (2020) — link
  115. Value-Difference Based Exploration: Adaptive Control between Epsilon-Greedy and Softmax (2011)
  116. Optimal Transport (2009) — link
  117. Some better bounds on the variance with applications (2010) — link
  118. On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (1994) — link
  119. Discounted Robust Stochastic Games and an Application to Queueing Control (2011) — link
  120. Sur l’existence d’une plus petite sous-tribu exhaustive par paire (1978) — link
  121. Second order reflected backward stochastic differential equations (2013) — link
  122. Superhedging and Dynamic Risk Measures under Volatility Uncertainty (2012) — link
  123. Wellposedness of Second Order Backward SDEs (2012)
  124. Dual formulation of second order target problems (2013) — link
  125. Existence of equilibria in economies with infinitely many commodities (1972) — link
  126. Charges as equilibrium prices and asset bubbles (1989) — link
  127. Bubbles as Payoffs at Infinity (1997) — link
  128. Asset Price Bubbles in Incomplete Markets (2010) — link
  129. Bubbles and Charges (1992) — link

0 of 137 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.