Data-Guided Mark-Law Selection for Positive Mean-Reverting Jump Diffusions: Threshold Inference and Rare-Event Prediction

Abstract This paper studies inference for a positive mean-reverting jump diffusion when jump sizes are not observed directly but inferred from increments crossing detection thresholds. Such selection changes the distribution of the retained marks and the effective observation time for jump arrivals. We combine a threshold-conditioned mark likelihood with a survival-adjusted intensity estimator and compare several positive mark laws, many expressible as finite common-rate gamma mixtures. We show that the smallest active gamma shape determines the leading order of sub-threshold jump probabilities. For Zeghdoudi marks the probability is quadratic in the threshold, and an explicit second-order expansion quantifies the downward bias of an ordinary likelihood fitted to truncated marks. These mark-specific results supplement, rather than restate, the standard affine properties of the underlying JCIR process. The paper also examines asymptotic inference under assumptions controlling jump-classification errors, parameter recovery in simulation, and threshold sensitivity in an annual Nile-flow illustration. The empirical ranking is stable over the reported threshold grid, although small information-criterion differences leave substantial mark-law uncertainty. MSC 2020: 62M05; 62F12; 60J60; 60G51; 62P12.