Distributional representations and dominance of a Lévy process over its maximal jump processes

Abstract

Distributional identities for a Lévy process Xt , its quadratic variation process Vt and its maximal jump processes, are derived, and used to make “small time” (as t ↓ 0) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of X. Apart from providing insight into the connections between X, V , and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of Xt , that is, Xt after division by sup0<s≤t Xs, or by sup0<s≤t |Xs|. Thus, we obtain necessary and sufficient conditions for Xt / sup0<s≤t Xs and Xt / sup0<s≤t |Xs| to converge in probability to 1, or to ∞, as t ↓ 0, so that X is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of X is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of X at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as t → ∞) versions of the results can also be obtained.