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Small deviations of general Lévy processes. (English) Zbl 1187.60035
Let \(X=(X_t)_{t\in[0,1]}\) be a real–valued Lévy process and let \((\nu,\sigma^2,b)\) be the corresponding Lévy triplet where \(b\in \mathbb R\), \(\sigma^2\geq 0\) and \(\nu\) denotes the generating Lévy measure. The authors describe the behavior of the small deviation function \[ -\log\mathbb P(\sup_{0\leq t\leq 1}|X_t|\leq \varepsilon) \] as \(\varepsilon\to 0\) in terms of quantities derived from \((\nu,\sigma^2,b)\). This is a very surprising result because, to our knowledge, this is the first example where one is able to describe the small deviation behavior for processes of a whole class. The paper contains several examples where the above mentioned quantities are calculated explicitly. Among them are Gamma processes, compound Poisson processes or the class of Lévy processes with non-zero Gaussian component. In the latter case the small deviation behavior coincides with that of the Brownian motion.

MSC:
60G51 Processes with independent increments; Lévy processes
60F99 Limit theorems in probability theory
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