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

Abstract

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) 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 $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, 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 $t\to\infty$) versions of the results can also be obtained.