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Exit times for a class of piecewise exponential Markov processes with two-sided jumps

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Title: Exit times for a class of piecewise exponential Markov processes with two-sided jumps
Authors: Jacobsen, Martin1 martin@math.ku.dk, Jensen, Anders Tolver2 tolver@kvl.dk
Source: Stochastic Processes & Their Applications. Sep2007, Vol. 117 Issue 9, p1330-1356. 27p.
Subjects: Markov processes, Probability theory, Eigenfunctions, Integrals
Abstract: Abstract: We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein–Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals. [Copyright &y& Elsevier]
Copyright of Stochastic Processes & Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 25937371
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  Data: Exit times for a class of piecewise exponential Markov processes with two-sided jumps
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  Data: <searchLink fieldCode="AR" term="%22Jacobsen%2C+Martin%22">Jacobsen, Martin</searchLink><relatesTo>1</relatesTo><i> martin@math.ku.dk</i><br /><searchLink fieldCode="AR" term="%22Jensen%2C+Anders+Tolver%22">Jensen, Anders Tolver</searchLink><relatesTo>2</relatesTo><i> tolver@kvl.dk</i>
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  Data: <searchLink fieldCode="JN" term="%22Stochastic+Processes+%26+Their+Applications%22">Stochastic Processes & Their Applications</searchLink>. Sep2007, Vol. 117 Issue 9, p1330-1356. 27p.
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  Data: <searchLink fieldCode="DE" term="%22Markov+processes%22">Markov processes</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+theory%22">Probability theory</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenfunctions%22">Eigenfunctions</searchLink><br /><searchLink fieldCode="DE" term="%22Integrals%22">Integrals</searchLink>
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  Label: Abstract
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  Data: Abstract: We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein–Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals. [Copyright &y& Elsevier]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Stochastic Processes & Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.spa.2007.01.005
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      – Code: eng
        Text: English
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        PageCount: 27
        StartPage: 1330
    Subjects:
      – SubjectFull: Markov processes
        Type: general
      – SubjectFull: Probability theory
        Type: general
      – SubjectFull: Eigenfunctions
        Type: general
      – SubjectFull: Integrals
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      – TitleFull: Exit times for a class of piecewise exponential Markov processes with two-sided jumps
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              Text: Sep2007
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