Paper: arXiv 2611.07083

Authors: Erhan Bayraktar, Hao Xing

Abstract

We give necessary and sufficient coefficient conditions for stochastic exponentials driven by multidimensional diffusions to be true martingales. The classical reduction to non-exit of the diffusion with Girsanov-shifted drift separates the analysis into smooth boundary points, infinity, and nonsmooth boundary pieces. On smooth domains, under regularity and non-explosion hypotheses, martingality is equivalent to nonpositivity of the shifted Fichera drift, provided an explicit local condition holds where that drift vanishes and the stochastic integrand is unbounded. At infinity, Feller’s test gives exact criteria when a scalar function of the diffusion reduces to a one-dimensional diffusion up to a time change with rate bounded above and away from zero. General radial and projection tests supply sufficient conditions. Applications give explicit necessary and sufficient conditions for inverse-CIR and $4/2$ models with correlated factors and singular stochastic integrands, including critical equality cases. The coupled non-affine examples go beyond direct scalar Feller tests and affine Riccati criteria, and exhibit boundary exits not decided by tests at infinity. Further exact criteria cover logarithmic boundary singularities, weighted distances, and an absorbed factor. The martingale conditions also yield uniqueness criteria for associated valuation equations.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper exhibits extremely high mathematical complexity, delving into rigorous proofs for martingale properties of multi-dimensional diffusions. While it provides theoretical applications to finance models, it lacks empirical backtesting or data validation, focusing purely on theoretical conditions. The novelty is high due to the derivation of necessary and sufficient conditions for complex, non-affine models.

Research Flowchart

  flowchart TD
    A[Research Goal: Necessary and sufficient conditions for Martingale property of Stochastic Exponentials] --> B{Key Methodology: Girsanov-shifted drift analysis, Feller's test};
    B --> C[Inputs: Coefficients of multi-dimensional diffusions, Stochastic integrands];
    C --> D{Computational Processes: Analysis at smooth boundaries, infinity, and non-smooth boundary pieces};
    D --> E[Key Findings: Coefficient conditions for martingality (nonpositivity of shifted Fichera drift, local conditions)];
    E --> F[Applications: Inverse-CIR, 4/2 models with correlated factors, logarithmic singularities];
    F --> G[Outcomes: Uniqueness criteria for associated valuation equations, exact criteria for critical cases];