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Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. (English) Zbl 1121.60029
Consider a $$d$$-dimensional Lévy process $$X= (X^1_t,\dots, X^d_t)_{0\leq t\leq 1}$$, which is regularly varying with index $$\alpha> 0$$, and a càdlàg predictable stochastic process $$Y= (Y^1_t,\dots, Y^d_t)_{0\leq t\leq 1}$$, which satisfies the momentum condition $$\mathbb{E}[\sup_{0\leq t\leq 1}\| Y_t\|^{\alpha+\varepsilon}]< \infty$$, for some $$\varepsilon> 0$$.
It is known that the extremal behaviour of $$X$$ is due to one large jump. The aim of this work is to establish that the same holds for the stochastic integral $(Y\cdot X)_{0\leq t\leq 1}:= \Biggl(\int^t_0 Y^1_s \,dX^1_s,\dots, \int^t_0 Y^d_s \,dX^d_s\Biggr)_{0\leq t\leq 1}.$ Precisely, the main result states that $\lim_{n\to\infty}\,\mathbb{P}\Biggl[\delta\Biggl({1\over n} Y\cdot X,{1\over n} Y_\tau\Delta X_\tau 1_{[\tau, 1]}\Biggr)> \varepsilon/\| Y\cdot X\|_\infty> n\Biggr]= 0$ and $\lim_{n\to\infty}\, \mathbb{P}\Biggl[\delta\Biggl({1\over n} Y\cdot X,{1\over n} Y_\tau\Delta X_\tau 1_{[\tau, 1]}\Biggr)> \varepsilon/|Y_\tau\Delta X_\tau|> n\Biggr]= 0,$ where $$\delta$$ denotes a metric on the space of càdlàg trajectories, and $$\tau$$ the time of the largest jump of $$X$$. Moreover, it is also shown that the stochastic integral $$Y\cdot X$$ is regularly varying with index $$\alpha$$, too, and its limit measure is calculated.

##### MSC:
 60F17 Functional limit theorems; invariance principles 60G17 Sample path properties 60H05 Stochastic integrals 60G70 Extreme value theory; extremal stochastic processes 60J75 Jump processes (MSC2010)
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