On the distribution of duration of stay in an interval of the semi-continuous process with independent increments

Kadankov, V. F.; Kadankova, Tetyana

doi:10.1515/1569397042722355

Cited by 9 publications

(16 citation statements)

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“…Other equalities can be verified analogously. Note, that the probabilities which enter the right-hand sides of the formulae of the corollary are the distributions of the number of the intersections of the interval (−r, k), k ∈ (0, 1), r = 1 − k by the Wiener process w t on the time interval [0, t] (see [13]). …”

Section: Asymptotic Resultsmentioning

confidence: 99%

“…The approach used for determining the Laplace transforms of the one-boundary characteristics of the process under consideration is based on the factorization methods. For determining the twoboundary characteristics of the process, we will follow the approach suggested in [13], [14] for Lévy processes. For convenience, we will use the same notation as in [18].…”

Section: Introductionmentioning

confidence: 99%

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Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

2009

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In this article we determine the Laplace transforms of the one-boundary characteristics and of the distribution of the number of intersections of a fixed interval by a difference of a compound Poisson process and a compound renewal process. The results obtained are applied for a particular case of this process, namely, for the difference of the compound Poisson process and the renewal process whose jumps are geometrically distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. In this case, under certain assumptions, we find the limit distributions of the one-boundary and two-boundary characteristics of the process. In addition, we prove the weak convergence of these distributions to the corresponding distributions of a symmetric Wiener process.

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Section: Asymptotic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

2009

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“…This allows one to apply these results to solve other two-boundary problems (see [17]- [19] where a number of similar problems are solved for semi-continuous stochastic processes with independent increments).…”

Section: Poisson Process With An Exponential Component 25mentioning

confidence: 99%

“…The joint distribution of the number of intersections of an interval is obtained in [18] for lower semi-continuous stochastic processes with independent increments. A limit theorem is proved in [18] on the weak convergence of the joint distribution of the numbers of intersections of an interval by a semi-continuous process (if the space and time are normalized suitably) to the joint distribution of the numbers of intersections of an interval by a symmetric Wiener process obtained in [22,18].…”

Section: Intersections Of An Interval By a Poisson Processmentioning

confidence: 99%

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

Kadankova

2008

Theor. Probability and Math. Statist.

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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.

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“…Kadankov and Kadankova [8] have used probabilistic methods (the total probability law, space homogeneity and the strong Markov property of the process) to determine the integral transforms E[e −sχ(y) ; X(y) ∈ du, A B ], E[e −sχ(y) ; X(y) ∈ du, A 0 ] of the joint distribution of {χ(y), X(y)}. For a spectrally positive Lévy process several two-boundary problems have been solved in [9]- [11].…”

Section: Introductionmentioning

confidence: 99%

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

Kadankova

Veraverbeke

2007

J Theor Probab

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Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms. IntroductionWe assume that all random variables and stochastic processes are defined on (Ω, F, {F t }, P ), a filtered probability space, where the filtration {F t } satisfies the usual conditions of rightcontinuity and completion. A Lévy process is a F -adapted stochastic process {ξ(t); t ≥ 0} which has independent and stationary increments and whose paths are right-continuous with left limits [1]. Under the assumption that ξ(0) = 0 the Laplace transform of the process {ξ(t); t ≥ 0} has the form E[e −pξ(t) ] = e t k(p) , Re p = 0, where the function k(p) is called the Laplace exponent and is given by the formula ([6])Here α, σ ∈ R and Π(·) is a measure on the real line. The introduced process is a space homogeneous, strong Markov process. Note, that the distribution of the first exit time from an interval plays a crucial role in applications and its knowledge also allows to solve a number of other two-boundary problems. Let us fix B > 0 and define the variable

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On the distribution of duration of stay in an interval of the semi-continuous process with independent increments

Cited by 9 publications

References 1 publication

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

Intersections of an Interval By a Difference of a Compound Poisson Process and a Compound Renewal Process

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

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

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