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Exit properties of stochastic processes with stationary independent increments


Author: P. W. Millar
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0321198-8
Keywords: Stochastic process, Markov process, stationary independent increments, sample function behavior, local time, Lévy measure, first passage time, first passage distribution, subordinator
Article copyright: © Copyright 1973 American Mathematical Society

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