Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae
Jacek Jakubowski (University of Warsaw and Warsaw University of Technology)
Mariusz Niewęgłowski (Warsaw University of Technology)
Abstract
In this paper we examine the problem of existence and construction of multivariate Markov chains such that their components are Markov chains with given laws. Specifically, we provide sufficient and necessary conditions, in terms of semimartingale characteristics, for a component of a multivariate Markov chain to be a Markov chain in its own filtration - a property called weak Markov consistency. Accordingly, we introduce and discuss the concept of weak Markov copulae. Finally, we examine relationship between the concepts of weak Markov consistency and weak Markov copulae, and the corresponding strong versions of these concepts.
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Pages: 1-21
Publication Date: March 31, 2013
DOI: 10.1214/EJP.v18-2238
References
- Ball, Frank; Yeo, Geoffrey F. Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Probab. 30 (1993), no. 3, 518--528. MR1232731
- Bielecki, T.R., Cousin, A., Crépey, S. and Herbertsson, A., Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model, Journal of Optimization Theory and Applications, forthcoming (2013).
- Bielecki, T.R., Vidozzi, A., Vidozzi, L., Multivariate Markov Processes with Given Markovian Margins, and Markov Copulae. preprint, Illinois Insitute of Technology (2007).
- Bielecki, T.R., Vidozzi, A., Vidozzi, L., A Markov Copulae Approach to Pricing and Hedging of Credit Index Derivatives and Ratings Triggered Step--Up Bonds. Journal of Credit Risk 4 No. 1, (2008), 47--76.
- Bielecki, Tomasz R.; Jakubowski, Jacek; Vidozzi, Andrea; Vidozzi, Luca. Study of dependence for some stochastic processes. Stoch. Anal. Appl. 26 (2008), no. 4, 903--924. MR2431085
- Bielecki, T. R. and Jakubowski, J. and Nieweglowski, M., Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes. Copula Theory and Its Applications, Lecture Notes in Statistics 198, Part 1, (2010), 33--76.
- Bielecki, Tomasz R.; Jakubowski, Jacek; Niewȩgłowski, Mariusz. Study of dependence for some stochastic processes: symbolic Markov copulae. Stochastic Process. Appl. 122 (2012), no. 3, 930--951. MR2891442
- Burke, C. J.; Rosenblatt, M. A Markovian function of a Markov chain. Ann. Math. Statist. 29 (1958) 1112--1122. MR0101557
- Chen, Mu-Fa. From Markov chains to non-equilibrium particle systems. Second edition. World Scientific Publishing Co., Inc., River Edge, NJ, 2004. xii+597 pp. ISBN: 981-238-811-7 MR2091955
- Jeanblanc, Monique; Yor, Marc; Chesney, Marc. Mathematical methods for financial markets. Springer Finance. Springer-Verlag London, Ltd., London, 2009. xxvi+732 pp. ISBN: 978-1-85233-376-8 MR2568861
- Dynkin, E.B., Markov Processes, Volume I. Springer--Verlag, 1965.
- Elliott, R.J. and Aggoun, L. and Moore, J.B., Hidden Markov Models: Estimation and Control. Springer, Berlin Heidelberg New York, 1994.
- Nelsen, Roger B. An introduction to copulas. Second edition. Springer Series in Statistics. Springer, New York, 2006. xiv+269 pp. ISBN: 978-0387-28659-4; 0-387-28659-4 MR2197664
- Rogers, L. C. G.; Pitman, J. W. Markov functions. Ann. Probab. 9 (1981), no. 4, 573--582. MR0624684
- Rogers, L.C.G. and Williams D., Diffusions, Markov Processes and Martingales. Cambridge University Press, 2000. MR1780932
- Ryan, Raymond A. Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2002. xiv+225 pp. ISBN: 1-85233-437-1 MR1888309
- Sklar, Abe. Random variables, joint distribution functions, and copulas. Kybernetika (Prague) 9 (1973), 449--460. MR0345164
- Vidozzi, A., Two Essays In Mathematical Finance. Doctoral Dissertation, Illinois Institute of Technology, 2009.
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