Hostname: page-component-76d6cb85b7-hqrjx Total loading time: 0 Render date: 2026-07-17T13:25:25.046Z Has data issue: false hasContentIssue false
  • English
  • Français

The impact of pinning points on memorylessness in Lévy random bridges

Part of: Markov processes Stochastic processes Distribution theory - Probability

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: m.louriki@uca.ac.ma
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 172 - 187
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bedini, M. L., Buckdahn, R. and Engelbert, H. J. (2017). Brownian bridges on random intervals. Theory Prob. Appl. 61, 15–39.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2007). Beyond hazard rates: A new framework to credit-risk modelling. In Advances in Mathematical Finance, eds M. Fu, R. Jarrow, J.-Y. J. Yen and R. Elliott, Birkhäuser, Basel, pp. 231–257.CrossRefGoogle Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2008). Information-based asset pricing. Internat. J. Theor. Appl. Finance 11, 107–142.CrossRefGoogle Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2008). Dam rain and cumulative gain. Proc. R. Soc. Lond. A 464, 1801–1822.CrossRefGoogle Scholar
Brody, D. C. and Law, Y. T. (2015). Pricing of defaultable bonds with random information flow. Appl. Math. Finance 22, 399–420.CrossRefGoogle Scholar
Chaumont, L. and Uribe-Bravo, G. (2011). Markovian bridges: Weak continuity and pathwise constructions. Ann. Prob. 39, 609–647.CrossRefGoogle Scholar
Erraoui, M. and Louriki, M. (2018). Bridges with random length: Gaussian-Markovian case. Markov Process. Relat. Fields 4, 669–693.Google Scholar
Erraoui, M., Hilbert, A. and Louriki, M. (2020). Bridges with random length: Gamma case. J. Theoret. Prob. 33, 931–953.CrossRefGoogle Scholar
Erraoui, M., Hilbert, A. and Louriki, M. (2021). On a Lévy process pinned at random time. Forum Math. 33, 397–417.CrossRefGoogle Scholar
Erraoui, M., Hilbert, A., and Louriki, M. (2021). On an extension of the Brownian bridge with some applications in finance. Preprint, arXiv:2110.01316.Google Scholar
Fitzsimmons, P. J., Pitman, J. and Yor, M. (1993). Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes (Progress in Prob. 33), eds E. Çinlar, K. L. Chung, M. J. Sharpe, R. F. Bass, and K. Burdzy, Springer, New York, 102–133.CrossRefGoogle Scholar
Hartman, P. and Wintner, A. (1942). On the infinitesimal generators of integral convolutions. Amer. J. Math. 64, 273–298.CrossRefGoogle Scholar
Hoyle, E., Hughston, L. P. and Macrina, A. (2011). Lévy random bridges and the modelling of financial information. Stoch. Process. Appl. 121, 856–884.CrossRefGoogle Scholar
Hoyle, E., Hughston, L. P. and Macrina, A. (2015). Stable- $\frac 12$ bridges and insurance. In Advances in Mathematics of Finance, eds A. Palczewski and L. Stettner, Banach Center Publications, Warsaw, pp. 95–120.Google Scholar
Louriki, M. (2022). Brownian bridge with random length and pinning point for modelling of financial information. Stochastics 94, 973–1002.CrossRefGoogle Scholar
Louriki, M. (2024). Information-based approach: Pricing of a credit risky asset in the presence of default time. Prob. Uncertain. Quant. Risk 9, 405–430.CrossRefGoogle Scholar
Rutkowski, M and Yu, N. (2007). An extension of the Brody–Hughston–Macrina approach to modelling of defaultable bonds. Internat. J. Theor. Appl. Finance 10, 557–589.CrossRefGoogle Scholar
Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sharpe, M. (1969). Zeroes of infinitely divisible densities. Ann. Math. Statist. 40, 1503–1505.CrossRefGoogle Scholar
Tucker, H. G. (1962). Absolute continuity of infinitely divisible distributions. Pacific J. Math. 12, 1125–1129.CrossRefGoogle Scholar