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Separating Times for One-Dimensional General Diffusions

David Criens, Mikhail Urusov

TL;DR

The paper investigates separating times for laws of general one-dimensional diffusions, providing a representation of the separating time as a hitting-time type object characterized by the diffusion's scale function and speed measure.A sequence of deterministic conditions is derived that determine when two laws are locally absolutely continuous or singular, taking into account boundary accessibility, stickiness, and instantaneous or slow reflection.These results extend the Itô-diffusion literature by accommodating general speeds/scales and boundary behaviors, with applications to financial modeling where NFLVR and the unique equivalent local martingale measure are determined deterministically in a general diffusion framework.The approach combines time-change and symmetrization techniques with a detailed analysis of separating points, yielding explicit, checkable criteria and illuminating the role of boundary behavior in absolute continuity and arbitrage considerations.

Abstract

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.

Separating Times for One-Dimensional General Diffusions

TL;DR

The paper investigates separating times for laws of general one-dimensional diffusions, providing a representation of the separating time as a hitting-time type object characterized by the diffusion's scale function and speed measure.A sequence of deterministic conditions is derived that determine when two laws are locally absolutely continuous or singular, taking into account boundary accessibility, stickiness, and instantaneous or slow reflection.These results extend the Itô-diffusion literature by accommodating general speeds/scales and boundary behaviors, with applications to financial modeling where NFLVR and the unique equivalent local martingale measure are determined deterministically in a general diffusion framework.The approach combines time-change and symmetrization techniques with a detailed analysis of separating points, yielding explicit, checkable criteria and illuminating the role of boundary behavior in absolute continuity and arbitrage considerations.

Abstract

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.
Paper Structure (26 sections, 69 theorems, 299 equations, 1 figure)

This paper contains 26 sections, 69 theorems, 299 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Separating Times for General Diffusions
  3. Separating Times: Encoding Absolute Continuity and Singularity
  4. A Recap on the Canonical Diffusion Framework
  5. Main Results
  6. Deterministic Conditions for NFLVR in General Diffusion Models
  7. Proof of the Main Theorem
  8. Some Preparations
  9. Proof of Lemma \ref{['lem:130222a2']}
  10. Criteria for Equivalence
  11. Singularity under time transformations
  12. Criteria for Singularity
  13. Proof of Theorem \ref{['theo: main1']}
  14. Examples for Separating Times
  15. Martingale Problem for Diffusions
  16. ...and 11 more sections

Key Result

Theorem 2.2

There exists a $\mathds{P}, \tilde{\mathds{P}}$-a.s. unique extended stopping time $S$ such that for any stopping time $\tau$ we have This extended stopping time is called the separating time for $\mathds{P}$ and $\tilde{\mathds{P}}$.

Figures (1)

  • Figure 1: Share Price Hansen Technologies Limited (HSN.AX)

Theorems & Definitions (131)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6: Non-separating or good interior point
  • Remark 2.7
  • Lemma 2.9: Equivalent definition of a non-separating interior point
  • Remark 2.10
  • proof : Proof of Lemma \ref{['lem:130222a1']}
  • ...and 121 more