Abstract
Let be a Hunt process with state space (E, ℰ). In [4] Weil proved the conditional independence of and θT given XT _ for certain hitting times T of the process (Xt _,Xt)t>0. In [3] Pitman investigated the stochastic dependence of and θT for arbitrary stopping times T. Here we shall look at this question for arbitrary times T. The main result is formula (3) below which contains both Weil’s result, its extension to arbitrary hitting times of (Xt_,Xt) (Theorem 2) and a result for last exit times. This was thought to be an introduction to similar formulae for the excursions straddling arbitrary times, but we had no time to work out the details for the present publication.
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References
A. Benveniste and J. Jacod. Systèmes de Lévy des processus de Markov. Invent. Math. 21 (1973), 183–198.
C. Dellacherie. Au sujet des sauts d’un processus de Hunt. Séminaire de Probabilités IV, 71–72, Lecture Notes Math. 124. Springer, Berlin, 1970.
J. W. Pitman. Lévy Systems and path decompositions. Seminar on Stochastic Processes, 1981, 79–110. Birkhäuser, Boston, 1981.
M. Weil. Conditionnement par rapport au passé strict. Séminaire de Probabilités V, 362–372, Lecture Notes Math. 191. Springer, Berlin, 1971.
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© 1986 Birkhäuser Boston, Inc.
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Maisonneuve, B. (1986). Strict Past Conditioning at Arbitrary Times. In: Çinlar, E., Chung, K.L., Getoor, R.K., Glover, J. (eds) Seminar on Stochastic Processes, 1985. Progress in Probability and Statistics, vol 12. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6748-2_10
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DOI: https://doi.org/10.1007/978-1-4684-6748-2_10
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