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Probability of total domination for transient reflecting processes in a quadrant

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  14 June 2022

Vladimir Fomichov Open the ORCID record for Vladimir Fomichov [Opens in a new window] ,
Sandro Franceschi Open the ORCID record for Sandro Franceschi [Opens in a new window]  and
Jevgenijs Ivanovs
Vladimir Fomichov*
Affiliation:
Aarhus University
Sandro Franceschi*
Affiliation:
Télécom SudParis
Jevgenijs Ivanovs*
Affiliation:
Aarhus University
*
*Postal address: Ny Munkegade 118, Aarhus University, 8000 Aarhus, Denmark.
***Postal address: 19 place Marguerite Perey, Télécom SudParis, Bâtiment A, 91120 Palaiseau, France. Email address: sandro.franceschi@telecom-sudparis.eu
*Postal address: Ny Munkegade 118, Aarhus University, 8000 Aarhus, Denmark.
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Abstract

We consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that such behaviour is certain and each component can dominate the other with positive probability for any given starting position. Additionally, we establish the corresponding invariance principle providing justification for the use of the reflected Brownian motion as an approximate model. Focusing on the probability that the first component dominates, we derive a kernel equation for the respective Laplace transform in the starting position. This is done for the compound Poisson model with negative exponential jumps and, by means of approximation, for the Brownian model. Both equations are solved via boundary value problem analysis, which also yields the domination probability when starting at the origin. Finally, certain asymptotic analysis and numerical results are presented.

Keywords

Carleman boundary value problemkernel equationLévy processesreflected Brownian motionSkorokhod problemuniform law of large numbers

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60J65: Brownian motion
Secondary: 60K25: Queueing theory 60K40: Other physical applications of random processes
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1094 - 1138
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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