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1 2021

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On the Exact Asymptotics of Exit Time From a Cone of an Isotropic α-Self-Similar Markov Process with a Skew-Product Structure

Palmowski Zbigniew, Wang Longmin

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

25--38

In this paper we identify the asymptotic tail of the distribution of the exit time τC from a cone C of an isotropic α-self-similar Markov process Xt with a skew-product structure, that is, Xt is a product of its radial process and an independent time changed angular component θt. Under some additional regularity assumptions, the angular process θt killed on exiting the cone C has a transition density that can be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed Lévy process related to the Lamperti representation of the radial process, we prove that Px(τC> t) ~h(x)t-k1 as t→∞ for h and k1 identified explicitly. The result extends the work of De Blassie (1988) and Bañuelos and Smits (1997) concerning the Brownian motion.