Zhan Shi

Summary: We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. 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.

Keywords: Lévy process, subordination, small deviation, fractional Brownian motion.

2000 Mathematics Subject Classification: 60G51, 60G15, 60G52.

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