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The impact of pinning points on memorylessness in Lévy random bridges

Part of: Distribution theory - Probability Markov processes Stochastic processes

Published online by Cambridge University Press:  15 October 2024

Mohammed Louriki
Mohammed Louriki*
Affiliation:
Cadi Ayyad University
*
*Postal address: Mathematics Department, Faculty of Sciences Semalalia, Cadi Ayyad University, Boulevard Prince Moulay Abdellah, P. O. Box 2390, Marrakesh 40000, Morocco. Email: [email protected]
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Abstract

Random bridges have gained significant attention in recent years due to their potential applications in various areas, particularly in information-based asset pricing models. This paper aims to explore the potential influence of the pinning point’s distribution on the memorylessness and stochastic dynamics of the bridge process. We introduce Lévy bridges with random length and random pinning points, and analyze their Markov property. Our study demonstrates that the Markov property of Lévy bridges depends on the nature of the distribution of their pinning points. The law of any random variables can be decomposed into singular continuous, discrete, and absolutely continuous parts with respect to the Lebesgue measure (Lebesgue’s decomposition theorem). We show that the Markov property holds when the pinning points’ law does not have an absolutely continuous part. Conversely, the Lévy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.

Keywords

Lévy processesLévy bridgesMarkov processes

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 172 - 187
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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