## Exit properties of stochastic processes with stationary independent increments

HTML articles powered by AMS MathViewer

- by P. W. Millar PDF
- Trans. Amer. Math. Soc.
**178**(1973), 459-479 Request permission

## Abstract:

Let $\{ {X_t},t \geq 0\}$ be a real stochastic process with stationary independent increments. For $x > 0$, define the exit time ${T_x}$ from the interval $( - \infty ,x]$ by ${T_x} = \inf \{ t > 0:{X_t} > x\}$. A reasonably complete solution is given to the problem of deciding precisely when ${P^0}\{ {X_{{T_x}}} = x\} > 0$ and precisely when ${P^0}\{ {X_{{T_x}}} = x\} = 0$. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of ${X_t}$. A few applications of this result are discussed.## References

- R. M. Blumenthal and R. K. Getoor,
*Sample functions of stochastic processes with stationary independent increments*, J. Math. Mech.**10**(1961), 493–516. MR**0123362** - R. M. Blumenthal and R. K. Getoor,
*Local times for Markov processes*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**3**(1964), 50–74. MR**165569**, DOI 10.1007/BF00531683 - R. M. Blumenthal and R. K. Getoor,
*Markov processes and potential theory*, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR**0264757** - Salomon Bochner,
*Harmonic analysis and the theory of probability*, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR**0072370**, DOI 10.1525/9780520345294 - Leo Breiman,
*Probability*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR**0229267** - Jean Bretagnolle,
*Résultats de Kesten sur les processus à accroissements indépendants*, Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969-1970), Lecture Notes in Math., Vol. 191, Springer, Berlin, 1971, pp. 21–36 (French). MR**0368175** - Kai Lai Chung,
*A course in probability theory*, Harcourt, Brace & World, Inc., New York, 1968. MR**0229268**
B. E. Fristedt, Forthcoming monograph, Chap. 9.
- R. K. Getoor and H. Kesten,
*Continuity of local times for Markov processes*, Compositio Math.**24**(1972), 277–303. MR**310977** - Nobuyuki Ikeda and Shinzo Watanabe,
*On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes*, J. Math. Kyoto Univ.**2**(1962), 79–95. MR**142153**, DOI 10.1215/kjm/1250524975 - Harry Kesten,
*Hitting probabilities of single points for processes with stationary independent increments*, Memoirs of the American Mathematical Society, No. 93, American Mathematical Society, Providence, R.I., 1969. MR**0272059**
—, - John Lamperti,
*An invariance principle in renewal theory*, Ann. Math. Statist.**33**(1962), 685–696. MR**137176**, DOI 10.1214/aoms/1177704590 - P.-A. Meyer,
*Intégrales stochastiques. I, II, III, IV*, Séminaire de Probabilités (Univ. Strasbourg, Strasbourg, 1966/67) Springer, Berlin, 1967, pp. 72–94, 95–117, 118–141, 142–162 (French). MR**0231445** - P. A. Meyer,
*Un théorème sur la répartition des temps locaux*, Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969–1970), Lecture Notes in Math., Vol. 191, Springer, Berlin, 1971, pp. 209–210. MR**0400419** - Paul-André Meyer,
*Processus de Markov*, Lecture Notes in Mathematics, No. 26, Springer-Verlag, Berlin-New York, 1967 (French). MR**0219136** - P. W. Millar,
*Path behavior of processes with stationary independent increments*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**17**(1971), 53–73. MR**324781**, DOI 10.1007/BF00538475 - P. Warwick Millar,
*Stochastic integrals and processes with stationary independent increments*, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 307–331. MR**0402922** - B. A. Rogozin,
*Distribution of certain functionals related to boundary value problems for processes with independent increments*, Teor. Verojatnost. i Primenen.**11**(1966), 656–670 (Russian, with English summary). MR**0208682** - B. A. Rogozin,
*The local behavior of processes with independent increments*, Teor. Verojatnost. i Primenen.**13**(1968), 507–512 (Russian, with English summary). MR**0242261** - E. S. Štatland,
*On local properties of processes with independent increments*, Teor. Verojatnost. i Primenen.**10**(1965), 344–350 (Russian, with English summary). MR**0183022** - Shinzo Watanabe,
*On stable processes with boundary conditions*, J. Math. Soc. Japan**14**(1962), 170–198. MR**144387**, DOI 10.2969/jmsj/01420170 - Shinzo Watanabe,
*On discontinuous additive functionals and Lévy measures of a Markov process*, Jpn. J. Math.**34**(1964), 53–70. MR**185675**, DOI 10.4099/jjm1924.34.0_{5}3

*Continuity of local times for Markov processes*, Preliminary manuscript whose final version was [9].

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 459-479 - MSC: Primary 60J30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0321198-8
- MathSciNet review: 0321198