Paper: arXiv 2410.09789

Authors: David Criens, Mikhail Urusov

Abstract

In a seminal paper, F. Delbaen and W. Schachermayer proved that the classical NA (“no arbitrage”) condition implies the existence of an “absolutely continuous local martingale measure” (ACLMM). It is known that in general the existence of an ACLMM alone is not sufficient for NA. In this paper we investigate how close these notions are for single asset general diffusion market models. We show that NA is equivalent to the existence of an ACLMM plus a mild regularity condition on the scale function and the absence of reflecting boundaries. For infinite time horizon scenarios, the regularity assumption and the requirement on the boundaries can be dropped, showing equivalence between NA and the existence of an ACLMM. By means of counterexamples, we show that our characterization of NA for finite time horizons is sharp in the sense that neither the regularity condition on the scale function nor the absence of reflecting boundaries can be dropped.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This is a purely theoretical paper in mathematical finance focusing on no-arbitrage theory and stochastic calculus for general diffusion models. It delivers precise mathematical characterizations and counterexamples to resolve an open theoretical question regarding ACLMMs, but contains no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Bridge gap between No Arbitrage 'NA' and ACLMM in general diffusion models] --> B[Input: Single asset general diffusion market models & scale function analysis]
    B --> C[Analyze Time Horizons: Finite vs. Infinite]
    C --> D[Test Conditions: Evaluate scale function regularity & boundary behaviors]
    D --> E[Construct Counterexamples: Test necessity of conditions in finite time]
    
    E --> F[Outcome 1: Finite Horizon - NA ⇔ ACLMM + Scale Regularity + No Reflecting Boundaries]
    E --> G[Outcome 2: Infinite Horizon - NA ⇔ ACLMM]

Cited References (24)

  1. Criteria for the absence of arbitrage in general diffusion markets — → 2306.11470
  2. Separating Times for One-Dimensional General Diffusions (2022) — link
  3. Probability (2019) — link
  4. A one-dimensional diffusion hits points fast (2015) — link
  5. On the Loss of the Semimartingale Property at the Hitting Time of a Level (2013) — link
  6. The Mathematics of Arbitrage (2006) — link
  7. Essentials of stochastic finance (1999) — link
  8. Handbook of Brownian Motion - Facts and Formulae (1996) — link
and 16 more
  1. The Existence of Absolutely Continuous Local Martingale Measures (1995) (1995) — link
  2. A Necessary and Sufficient Condition for Absence of Arbitrage with Tame Portfolios (1995) — link
  3. Equivalent martingale measures and no-arbitrage in stochastic securities market models (1990) — link
  4. Martingales and stochastic integrals in the theory of continuous trading (1981) — link
  5. On Skew Brownian Motion (1981) — link
  6. Martingales and arbitrage in multiperiod securities markets (1979) — link
  7. Foundations of Modern Probability (2021) — link
  8. Markov Processes and Martingales (2016)
  9. Brownian Motion and Stochastic Calculus (2008) — link
  10. Remarks on the true No-arbitrage Property (2005) — link
  11. Continuous martingales and Brownian motion (1990) — link
  12. Strong Markov Continuous Local Martingales and Solutions of One‐Dimensional Stochastic Differential Equations (Part III) (1989) — link
  13. Limit Theorems for Stochastic Processes (1987) — link
  14. Semimartingales and Markov processes (1980) — link
  15. Calcul stochastique et problèmes de martingales (1979) — link
  16. Processes and their Sample Paths (1974)

1 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.