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Distributional Representations and Dominance of a Levy Process over its Maximal Jump Processes

Distributional Representations and Dominance of a Levy Process over its Maximal Jump Processes

Boris Buchmann, Yuguang Fan and Ross Maller

Distributional identities for a L evy process X, its quadratic variation process V, and its maximal jump processes sup 0<s \le t DX_s, are derived, and used to make (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 a self-normalized versions of X_t, that is, X_t after division by sup0<s \le t DX_s, or by sup0<s \le t |DX_s|. Thus we obtain necessary and sufficient conditions for X_t/ sup0<s\le t DX_s and X_t/sup 0<s\le t |DX_s| to converge in probability to 1, or infinity, 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 L evy 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 infinity) versions of the results can also be obtained.

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Updated:   18 February 2017 / Responsible Officer:  Dean, Business & Economics / Page Contact:  College Web Team