Necessary and sufficient conditions for the martingale property of stochastic exponentials driven by multi-dimensional diffusions
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.
Related
- Necessary and sufficient conditions for optimal control of coupled forward-backward stochastic control systems driven by G-Brownian motion
- Necessary and Sufficient Conditions for Adaptive Stabilizability of Continuous-Time Markov Jump Linear Stochastic Systems
- Form factors of field exponentials in the 1+1 dimensional Sine-Gordon model
- On the existence of a weak martingale solution for a stochastic magnetohydrodynamics system with noise acting in the magnetic field
- Càdlàg Solutions to Backward Stochastic Dynamics featuring Oblique Subgradients and driven by Martingale Noise
BibTeX
@misc{bayraktar2026necessary,
title = {{Necessary and sufficient conditions for the martingale property of stochastic exponentials driven by multi-dimensional diffusions}},
author = {Erhan Bayraktar and Hao Xing},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07083},
eprint = {2611.07083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07083},
}
RIS
TY - UNPB
TI - Necessary and sufficient conditions for the martingale property of stochastic exponentials driven by multi-dimensional diffusions
AU - Erhan Bayraktar
AU - Hao Xing
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07083
UR - https://arxiv.org/abs/2611.07083
AB - 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.
ER -