Exit, passage, and crossing times and overshoots for a Poisson compound process with an exponential component

Abstract

Abstract. Integral transforms of the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time are found for a Poisson process with an exponentially distributed negative component. We obtain the distributions of the following functionals of the process on an exponentially distributed time interval: the supremum, infimum, and the value of the process, numbers of upcrossings and downcrossings, the number of passages into an interval and overshoots over a boundary of an interval. IntroductionThe main two-boundary functional for homogeneous stochastic processes with independent increments is the joint distribution of the first exit time from an interval and overshoot over the boundary at the exit time. Below we give a short survey of results related to this functional. We do not pretend to cover all the aspects of the theory; the survey reflects an author personal viewpoint and approach to the above functional in the case of stochastic processes with independent increments.Let ξ(t) ∈ R, t ≥ 0, ξ(0) = 0, be a homogeneous stochastic process with independent increments [1] whose cumulant is given by (Re p = 0) We consider the random events A x = {ξ(χ) > x} and A y = {ξ(χ) < −y} meaning that the process exits the interval through the upper and lower boundary, respectively. Further letbe the overshoot of the process over the boundary at the first exit time from an interval where I A = I A (ω) is the indicator of a random event A.2000 Mathematics Subject Classification. Primary 60J05, 60J10; Secondary 60J45. Key words and phrases. Poisson process with an exponentially distributed negative component, oneboundary functionals of a process, exit times from an interval, overshoot over a boundary, supremum and infimum of the process, crossing times for an interval.

Multilayers in a modulated stochastic game

“…An interesting article [30] by Kadankova is one of the recent papers on the theory of fluctuations somewhat (but not directly) related to ours. In this paper the author derives the joint distribution of the first exit time from an interval and the excess over a threshold at the exit time for a Poisson process with an exponentially distributed negative component and the supremum, infimum, and the number of upcrossings and downcrossings, the number of passages into an exponentially distributed interval of time and the excess over a boundary of an interval.…”

On the Exiting Patterns of Multivariate Renewal-Reward Processes with an Application to Stochastic Networks

“…Then, notice that if we apply d operators of the appropriate types to 1 W , the result is a product of d terms similar to (21) or (22), which, after carefully handling the indices and notation, leads to (18).…”

“…More infrequent are studies of the threshold crossings of sums of independent random vectors, which may be considered in the context of multidimensional renewal processes [12,13] or random walks [14,15]. Most of these tend to focus on exit times [16][17][18][19][20] and overshoots of the boundary [21,22]. These latter ideas are commonly studied in the context of broader Lévy processes as well [23][24][25], but frequently only in one dimension.…”