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Evaluating the small deviation probabilities for subordinated Lévy processes

Author

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  • Linde, Werner
  • Shi, Zhan

Abstract

We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. These processes can be represented as WoA, where W is a standard Brownian motion, and A is a subordinator independent of W. Under some mild general assumption, we give precise estimates (up to a constant multiple in the logarithmic scale) of the small deviation probabilities. These probabilities, also evaluated under the conditional probability given the subordination process A, are formulated in terms of the Laplace exponent of A. The results are furthermore extended to processes subordinated to the fractional Brownian motion of arbitrary Hurst index.

Suggested Citation

  • Linde, Werner & Shi, Zhan, 2004. "Evaluating the small deviation probabilities for subordinated Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 273-287, October.
    • Handle: RePEc:eee:spapps:v:113:y:2004:i:2:p:273-287
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    Cited by:

    1. Aurzada, Frank & Lifshits, Mikhail, 2008. "Small deviation probability via chaining," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2344-2368, December.

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