Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T23:34:20.037Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised liouville processes and their properties

Part of: Markov processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  23 November 2020

Edward Hoyle  and
Levent Ali Menguturk Open the ORCID record for Levent Ali Menguturk [Opens in a new window]
Edward Hoyle*
Affiliation:
AHL Partners LLP
Levent Ali Menguturk*
Affiliation:
University College London
*
*Postal address: AHL Partners LLP, Man Group plc, London EC4R 3AD, UK.
**Postal address: Department of Mathematics, University College London, London WC1E 6BT, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a new family of multivariate stochastic processes over a finite time horizon that we call generalised Liouville processes (GLPs). GLPs are Markov processes constructed by splitting Lévy random bridges into non-overlapping subprocesses via time changes. We show that the terminal values and the increments of GLPs have generalised multivariate Liouville distributions, justifying their name. We provide various other properties of GLPs and some examples.

Keywords

Lévy processLévy bridgeMarkov processmultivariate Liouville distribution

MSC classification

Primary: 60G20: Generalized stochastic processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60H05: Stochastic integrals
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1088 - 1110
Check for updates
Copyright
© Applied Probability Trust 2020

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.)

References

Bielecki, T. R., Jakubowski, J. and Nieweglowski, M. (2017). Conditional Markov chains: properties, construction and structured dependence. Stoch. Process. Appl. 127, 1125–1170.10.1016/j.spa.2016.07.010CrossRefGoogle Scholar
Brody, D. C. and Hughston, L. P. (2005). Finite-time stochastic reduction models. J. Math. Phys. 46, 082101.10.1063/1.1990108CrossRefGoogle Scholar
Brody, D. C., Davis, M. H. A., Friedman, R. L. and Hughston, L. P. (2009). Informed traders. Proc. R. Soc. London A 465, 1103–1122.10.1098/rspa.2008.0465CrossRefGoogle Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2007). Beyond hazard rates: a new framework for credit-risk modelling. In Advances in Mathematical Finance, eds M. C. Fu et al., pp. 231–257. Birkhäuser, Boston.10.1007/978-0-8176-4545-8_13CrossRefGoogle Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2008). Dam rain and cumulative gain. Proc. R. Soc. London A 464, 1801–1822.10.1098/rspa.2007.0273CrossRefGoogle Scholar
Brody, D. C., Hughston, L. P. and Macrina, A. (2008). Information-based asset pricing. Int. J. Theor. Appl. Finance 11, 107–142.10.1142/S0219024908004749CrossRefGoogle Scholar
Chaumont, L., Hobson, D. G. and Yor, M. (2001). Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes. In Séminaire de Probabilités XXXV, eds J. Azéma et al., pp. 334–347. Springer, Berlin and Heidelberg.10.1007/978-3-540-44671-2_23CrossRefGoogle Scholar
Fang, K.-T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Chapman & Hall, New York.10.1007/978-1-4899-2937-2CrossRefGoogle Scholar
Fitzsimmons, P., Pitman, J. and Yor, M. (1992). Markovian bridges: construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, eds E. ×inlar et al., pp. 101–134. Birkhäuser, Boston.Google Scholar
Gupta, R. D. and Richards, D. St. P. (1987). Multivariate Liouville distributions. J. Multivariate Anal. 23, 233–256.10.1016/0047-259X(87)90155-2CrossRefGoogle Scholar
Gupta, R. D. and Richards, D. St. P. (1991). Multivariate Liouville distributions II. Prob. Math. Statist. 12, 291–309.Google Scholar
Gupta, R. D. and Richards, D. St. P. (1992). Multivariate Liouville distributions III. J. Multivariate Anal. 43, 29–57.10.1016/0047-259X(92)90109-SCrossRefGoogle Scholar
Gupta, R. D. and Richards, D. St. P. (1995). Multivariate Liouville distributions IV. J. Multivariate Anal. 54, 1–17.10.1006/jmva.1995.1042CrossRefGoogle Scholar
Hoyle, E. and Menguturk, L. A. (2013). Archimedean survival processes. J. Multivariate Anal. 115, 1–15.10.1016/j.jmva.2012.09.008CrossRefGoogle Scholar
Hoyle, E., Hughston, L. P. and Macrina, A. (2011). Lévy Random bridges and the modelling of financial information. Stoch. Process. Appl. 121, 856–884.10.1016/j.spa.2010.12.003CrossRefGoogle Scholar
Hoyle, E., Hughston, L. P. and Macrina, A. (2015). Stable-$1/2$ bridges and insurance. In Advances in Mathematics of Finance (Banach Center Publications 104), eds A. Palczewski and Ł. Stettner, pp. 95–120. Polish Academy of Sciences, Institute of Mathematics, Warsaw.Google Scholar
Jakubowski, J. and Pytel, A. (2016). The Markov consistency of Archimedean survival processes. J. Appl. Prob. 53, 392–409.10.1017/jpr.2016.8CrossRefGoogle Scholar
Mansuy, R. and Yor, M. (2005). Harnesses, Lévy bridges and Monsieur Jourdain. Stoch. Process. Appl. 115, 329–338.10.1016/j.spa.2004.09.001CrossRefGoogle Scholar