Evaluating the small deviation probabilities for subordinated Lévy processes

We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. These processes can be represented as W∘A, 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.