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Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

Part of: Stochastic processes

Published online by Cambridge University Press:  30 August 2022

Remco van der Hofstad Open the ORCID record for Remco van der Hofstad [Opens in a new window] ,
Stella Kapodistria ,
Zbigniew Palmowski Open the ORCID record for Zbigniew Palmowski [Opens in a new window]  and
Seva Shneer
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Stella Kapodistria*
Affiliation:
Eindhoven University of Technology
Zbigniew Palmowski*
Affiliation:
Wrocław University of Science and Technology
Seva Shneer*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
****Postal address: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland. Email: zbigniew.palmowski@pwr.edu.pl
*****Postal address: Heriot-Watt University, Edinburgh, UK. Email: v.shneer@hw.ac.uk
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Abstract

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

Keywords

Exit timesfirst passage timesadditive-increase and multiplicative-decrease processgrowth–collapse processLaplace–Stieltjes transformfirst-step analysisqueueing processstorageAIMD algorithm

MSC classification

Primary: 60G07: General theory of processes
Secondary: 60G44: Martingales with continuous parameter 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 85 - 105
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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