Summary
A Markov process of Ornstein-Uhlenbeck type was introduced in [5] as a Markov process onR d generated by a Lévy process generator plus a drift term\( - \sum\limits_{j = 1}^d {\sum\limits_{k = 1}^d {Q_{jk} } x_k \frac{\partial }{{\partial x_j }}}\) with the matrixQ=(Q jk) having eigenvalues with positive real parts. A criterion for positive recurrence of processes of this type was given by Sato-Yamazato [5]. This paper gives a criterion for null recurrence and transience by a integral condition involving the Lévy measure in the case of one dimension. Multi-dimensional cases are also discussed.
Article PDF
Similar content being viewed by others
References
Çinlar, E., Pinsky, M.: A stochastic integral in storage theory. Z. Wahrscheinlichkeitstheor. Verw. Geb.17, 227–240 (1971)
Hadjiev, D.I.: The first passage problem for generalized Ornstein Uhlenbeck processes with non-positive jumps (Lect. Notes Math., vol. 1123, pp. 80–90) Berlin Heidelberg New York: Springer 1985
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. New York Amsterdam: North-Holland/Kodansha 1981
Jurek, Z.J., Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 247–262 (1983)
Sato, K., Yamazato, M.: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Processes Appl.17, 73–100 (1984)
Sato, K.: Processes with independent increments. (in Japanese) Kinokuniya Company Limited (1990)
Sharpe, M.: General theory of Markov processes. New York London: Academic Press 1988
Wolfe, S.J.: On a continuous analogue of the stochastic difference equationX n =ρX n−1 +B n . Stochastic Processes Appl.12, 301–312 (1982)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shiga, T. A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type. Probab. Th. Rel. Fields 85, 425–447 (1990). https://doi.org/10.1007/BF01203163
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01203163