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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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