On weak notions of no-arbitrage in a 1D general diffusion market with interest rates

Paper: arXiv 2503.14078

Authors: Alexis Anagnostakis, David Criens, Mikhail Urusov

Abstract

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions “no increasing profit” (NIP), “no strong arbitrage” (NSA) and “no unbounded profit with bounded risk” (NUPBR) in one-dimensional general diffusion markets. These are markets with one risky asset, which is modeled as a regular continuous strong Markov process that is also a semimartingale, and a riskless asset that grows exponentially at a constant rate $r\in \mathbb{R}$. All deterministic criteria are provided in terms of the scale function and the speed measure of the risky asset process. Our study reveals a variety of surprising effects. For instance, irrespective of the interest rate, NIP is not excluded by reflecting boundaries or an irregular scale function. In the case of non-zero interest rates, it is even possible that NUPBR holds in the presence of reflecting boundaries and/or skew thresholds. In the zero interest rate regime, we also identify NSA as the minimal no arbitrage notion that excludes reflecting boundaries and that forces the scale function to be continuously differentiable with strictly positive absolutely continuous derivative, meaning that it is of the same form as for a stochastic differential equation.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers a highly rigorous, technical mathematical finance study establishing deterministic necessary and sufficient conditions for weak no-arbitrage notions in 1D diffusion markets. While abstract and devoid of empirical backtesting or data analysis, its theoretical contributions regarding scale functions, speed measures, and interest rate impacts are novel and mathematically complete.

Research Flowchart

  flowchart TD
    A[Goal: Characterize NIP, NSA, and NUPBR in 1D Diffusion Markets with Rate r] --> B[Inputs: Continuous Strong Markov Risky Asset X, Interest Rate r]
    B --> C[Compute Scale Function S x and Speed Measure m dx]
    C --> D[Analytical Evaluation of Boundary Conditions and Scale Regularity]
    D --> E[Key Finding: Deterministic Conditions for NIP, NSA, NUPBR via S x and m dx]
    D --> F[Key Finding: Under r ≠ 0, NUPBR holds despite Reflecting Boundaries/Skew Thresholds]
    D --> G[Key Finding: Under r = 0, NSA is Minimal Notion Excluding Reflecting Boundaries]

Cited References (34)

1. On the representation property for 1d general diffusion semimartingales (2024) — link
2. Separating Times for One-Dimensional General Diffusions (2022) — link
3. On skew sticky Brownian motion (2021) — link
4. No arbitrage in continuous financial markets (2018) — link
5. A one-dimensional diffusion hits points fast (2015) — link
6. Convergence of integral functionals of one-dimensional diffusions (2011) — link
7. Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models (2010) — link
8. M-6-On Minimal Market Models and Minimal Martingale Measures (2010) — link

0 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.