×

Copula-based Markov process. (English) Zbl 1435.62177

Summary: Starting from a bivariate copula family, this paper investigates the existence of a Markov process whose temporal dependence is modeled by the given copula family. Due to that the transition function plays a core role for constructing a Markov process, a transition function should be defined from a copula family. For this purpose, the modified partial Dini derivatives of a bivariate copula are defined and applied for defining transition probabilities, and some properties of the modified partial Dini derivatives are proved. A necessary and sufficient condition for the family of the defined transition probabilities to be a transition function is provided. Given a bivariate copula family, a sufficient condition for the existence of a Markov process is provided, where the Markov process has a transition function generated by the modified partial Dini derivatives of the bivariate copula family and the temporal dependence of the Markov process is modeled by the given copula family. The resulting Markov process is named as the copula-based Markov process. Moreover, under some assumptions the consistency of the bivariate copula family under the \(\ast\) product operation is necessary and sufficient for the existence of a Markov process. In terms of copulas, some criteria are provided for a copula-based Markov process to be path right-continuous with left limits or path continuous, and a necessary and sufficient condition for a time-homogeneous copula-based Markov process to be a Feller process is obtained. It is interesting that a Markov process with the transition function generated by the modified partial Dini derivatives of FGM copulas is not a Feller process. Finally, paths of some typical copula-based Markov processes are simulated to show the importance of fitting the copula method into the framework of stochastic processes.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62M02 Markov processes: hypothesis testing

Software:

QRM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmadi, S. S.; Gaillardetz, P., Modeling mortality and pricing life annuities with Lévy processes, Insurance Math. Econom., 64, 337-350 (2015) · Zbl 1348.62229
[2] Aït-Sahalia, Y., Telling from discrete data whether the underlying continuous-time model is a diffusion, J. Finance, 57, 2075-2112 (2002)
[3] Ballotta, L.; Haberman, S., The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case, Insurance Math. Econom., 38, 195-214 (2006) · Zbl 1101.60045
[4] Bassamboo, A.; Juneja, S.; Zeevi, A., Portfolio credit risk with extremal dependence: Asymptotic analysis and efficient simulation, Oper. Res., 56, 3, 593-606 (2008) · Zbl 1167.91362
[5] Beare, B. K., Copulas and temporal dependence, Econometrica, 78, 395-410 (2010) · Zbl 1202.91271
[6] Beare, B. K., Archimedean copulas and temporal dependence, Econometric Theory, 28, 6, 1165-1185 (2012) · Zbl 1281.62143
[7] Beare, B. K.; Seo, J., Time irreversible copula-based Markov models, Econometric Theory, 30, 5, 923-960 (2014) · Zbl 1314.62122
[8] Bibbona, E.; Sacerdote, L.; Torre, E., A copula-based method to build diffusion models with prescribed marginal and serial dependence, Methodol. Comput. Appl. Probab., 18, 3, 765-783 (2016) · Zbl 1351.60107
[9] Billingsley, P., Convergence of Probability Measures (1999), John Wiley & Sons: John Wiley & Sons New York · Zbl 0172.21201
[10] Brigo, D.; Pallavicini, A.; Torresetti, R., Credit Models and the Crisis: A Journey Into CDOs, Copulas, Correlations and Dynamic Models (2010), John Wiley & Sons
[11] Cherubini, U.; Gobbi, F.; Mulinacci, S., Convolution Copula Econometrics (2016), Springer · Zbl 1360.62006
[12] Cherubini, U.; Gobbi, F.; Mulinacci, S.; Romagnoli, S., Dynamic Copula Methods in Finance (2012), John Wiley & Sons: John Wiley & Sons Chichester
[13] Cherubini, U.; Luciano, E.; Vecchiato, W., Copula Methods in Finance (2004), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 1163.62081
[14] Choroś-Tomczyk, B.; Härdle, W. K.; Okhrin, O., Valuation of collateralized debt obligations with hierarchical Archimedean copulae, J. Empir. Financ., 24, 42-62 (2013)
[15] Christoffersen, P.; Errunza, V.; Jacobs, K.; Langlois, H., Is the potential for international diversification disappearing? A dynamic copula approach, Rev. Financ. Stud., 25, 12, 3711-3751 (2012)
[16] Darsow, W. F.; Nguyen, B.; Olsen, E. T., Copulas and Markov processes, Illinois J. Math., 36, 4, 600-642 (1992) · Zbl 0770.60019
[17] Das, S. R.; Duffie, D.; Kapadia, N.; Saita, L., Common failings: How corporate defaults are correlated, J. Finance, 62, 1, 93-117 (2007) · Zbl 1418.91584
[18] Durante, F.; Fernández-Sánchez, J.; Trutschnig, W., On the singular components of a copula, J. Appl. Probab., 52, 1175-1182 (2015) · Zbl 1336.60020
[19] Durante, F.; Jaworski, P., A new characterization of bivariate copulas, Comm. Statist. Theory Methods, 39, 16, 2901-2912 (2010) · Zbl 1203.62101
[20] Durante, F.; Klement, E. P.; Quesada-Molina, J.; Sarkoci, P., Remarks on two product-like constructions for copulas, Kybernetika, 43, 235-244 (2007) · Zbl 1136.60306
[21] Dynkin, E. B., Markov Processes, Vol. I (1965), Springer: Springer Berlin · Zbl 0132.37901
[22] Ethier, S. N.; Kurtz, T. G., Markov Processes: Characterization and Convergence (2009), John Wiley & Sons: John Wiley & Sons Hoboken
[23] Fernández-Sánchez, J.; Trutschnig, W., Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and iterated function systems, J. Theoret. Probab., 28, 1311-1366 (2015) · Zbl 1329.62269
[24] Fernández-Sánchez, J.; Trutschnig, W., Singularity aspects of Archimedean copulas, J. Math. Anal. Appl., 432, 103-113 (2015) · Zbl 1329.62247
[25] Fernández-Sánchez, J.; Trutschnig, W., Some members of the class of (quasi-) copulas with given diagonal from the Markov kernel perspective, Comm. Statist. Theory Methods, 45, 1508-1526 (2016) · Zbl 1337.62121
[26] Fernández-Sánchez, J.; Úbeda-Flores, M., Proving the characterization of Archimedean copulas via Dini derivatives, Kybernetika, 52, 5, 785-790 (2016) · Zbl 1389.62007
[27] Gulisashvili, A.; van Casteren, J. A., Non-Autonomous Kato Classes and Feynman-Kac Propagators (2006), World Scientific: World Scientific Singapore · Zbl 1113.47031
[28] Hull, J. C.; White, A. D., Valuing credit derivatives using an implied copula approach, J. Derivatives, 14, 2, 8-28 (2006)
[29] Ibragimov, R., Copula-Based Dependence Characteriztions and Modeling for Time SeriesHarvard Institute of Economic Research Discussion Paper No. 2094 (2005)
[30] Ibragimov, R., Copula-based characterizations for higher order Markov processes, Econometric Theory, 25, 3, 819-846 (2009) · Zbl 1277.60123
[31] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern Actuarial Risk Theory (2008), Springer: Springer Berlin · Zbl 1148.91027
[32] Kaas, R.; Laeven, R. J.; Nelsen, R. B., Worst VaR scenarios with given marginals and measures of association, Insurance Math. Econom., 44, 2, 146-158 (2009) · Zbl 1162.91417
[33] Kakouris, I.; Rustem, B., Robust portfolio optimization with copulas, European J. Oper. Res., 235, 1, 28-37 (2014) · Zbl 1305.91221
[34] Kallenberg, O., Foundations of Modern Probability (2002), Springer: Springer New York · Zbl 0996.60001
[35] Karoui, N. E.; Jeanblanc, M.; Jiao, Y., Density approach in modeling successive defaults, SIAM J. Financial Math., 6, 1, 1-21 (2015) · Zbl 1350.91018
[36] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality (2009), Birkhäuser: Birkhäuser Basel · Zbl 1221.39041
[37] Lagerås, A. N., Copulas for Markovian dependence, Bernoulli, 16, 2, 331-342 (2010) · Zbl 1323.60100
[38] Li, D. X., On default correlation: A copula function approach, J. Fixed Income, 9, 4, 43-54 (2000)
[39] Li, X.; Mikusiński, P.; Sherwood, H.; Taylor, M. D., On approximation of copulas, (Distributions with Given Marginals and Moment Problems (1997), Springer: Springer Dordrecht), 107-116 · Zbl 0905.60015
[40] Liggett, T. M., (Continuous Time Markov Processes: An Introduction. Continuous Time Markov Processes: An Introduction, Graduate studies in mathematics (2010), American Mathematical Society) · Zbl 1205.60002
[41] Łojasiewicz, S., An Introduction to the Theory of Real Functions (1988), John Wiley & Sons: John Wiley & Sons New York · Zbl 0653.26001
[42] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques, and Tools (2005), Princeton University Press · Zbl 1089.91037
[43] Nelsen, R. B., An Introduction to Copulas (2006), Springer Science & Business Media: Springer Science & Business Media New York · Zbl 1152.62030
[44] Oh, D. H.; Patton, A. J., Time-varying systemic risk: Evidence from a dynamic copula model of CDS spreads, J. Bus. Econom. Statist., 36, 2, 181-195 (2018)
[45] Olsen, E. T.; Darsow, W. F.; Nguyen, B., Copulas and Markov operators, (Proceedings of the Conference on Distributions with Fixed Marginals and Related Topics. Proceedings of the Conference on Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes, Monograph Series, vol. 28 (1996)), 244-259
[46] Overbeck, L.; Schmidt, W. M., Multivariate Markov families of copulas, Depend. Model., 3, 1, 159-171 (2015) · Zbl 1335.60134
[47] Patton, A. J., Modelling asymmetric exchange rate dependence, Internat. Econom. Rev., 47, 2, 527-556 (2006)
[48] Revuz, D.; Yor, M., Continuous Martingales and Brownian Motion (2013), Springer: Springer Berlin
[49] Trutschnig, W., On Cesáro convergence of iterates of the star product of copulas, Statist. Probab. Lett., 83, 357-365 (2013) · Zbl 1282.62148
[50] Trutschnig, W., Some smoothing properties of the star product of copulas, (Kruse, R.; Berthold, M.; Moewes, C.; Gil, M.; Grzegorzewski, P.; Hryniewicz, O., Synergies of Soft Computing and Statistics for Intelligent Data Analysis. Synergies of Soft Computing and Statistics for Intelligent Data Analysis, Advances in Intelligent and Soft Computing, vol. 190 (2013), Springer: Springer Heidelberg), 349-357 · Zbl 06580251
[51] Trutschnig, W.; Fernández-Sánchez, J., Idempotent and multivariate copulas with fractal support, J. Statist. Plann. Inference, 142, 3086-3096 (2012) · Zbl 1348.60020
[52] Trutschnig, W.; Schreyer, M.; Fernández-Sánchez, J., Mass distributions of two-dimensional extreme-value copulas and related results, Extremes, 19, 405-427 (2016) · Zbl 1359.62217
[53] Yang, J.; Cheng, S.; Zhang, L., Bivariate copula decomposition in terms of comonotonicity, countermonotonicity and independence, Insurance Math. Econom., 39, 2, 267-284 (2006) · Zbl 1098.62070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.