Separating Times and Phase Transitions in Non-Classical Diffusions: A Martingale Perspective

Criens & Urusov (2022) introduce separating times as extended stopping times marking the switch from equivalence to singularity between two probability measures on diffusions. Their work handles general scale and speed functions, uncovering novel phenomena not seen in SDEs with regular coefficients. This research idea proposes to deepen the martingale-theoretic analysis of these separating times: can we construct explicit martingales whose (almost sure) explosion or vanishing at the separating time signals a "phase transition"? Can we generalize optional sampling or Doob-Meyer-type results to this setting? This could provide new insights into change-of-measure problems, boundary behavior, and robust criteria for no-arbitrage in finance. The approach would also connect to the recent literature on invariance times (Cr'épey 2024), potentially uncovering a unified framework for "critical times" in stochastic models.

References:

1. Separating Times for One-Dimensional General Diffusions. D. Criens, M. Urusov (2022).
2. Invariance times transfer properties. St'ephane Cr'epey Ufr Math'ematiques UPCit'e, Lpsm (2024). Probability, Uncertainty and Quantitative Risk.