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2017 | Supplement | Chapter

A Note on Independence Copula for Conditional Markov Chains

Authors : Tomasz R. Bielecki, Jacek Jakubowski, Mariusz Niewęgłowski

Published in: Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Publisher: Springer New York

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Abstract

Given a family (Y ^k, k = 1, 2, …, N) of conditional Markov chains, we construct a conditional Markov chain X = (X ¹, …, X ^N) such that X ^k, k = 1, 2, …, N, are conditional Markov chains, which are conditionally independent given the information contained in some filtration \(\mathbb{F}\), and such that for each k the conditional law of X ^k coincides with the conditional law of Y ^k. This is a new result that can be used to model different phenomena such as the gating behavior of multiple ion channels in a membrane patch, or credit ratings migrations.

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previous chapter Calculation of Exposure Profiles and Sensitivities of Options under the Heston and the Heston Hull-White Models

next chapter The Construction and Properties of Assortative Configuration Graphs

In more generality, one might define strong Markovian consistency with respect to a collection X ^I: = {X ^k, k ∈ I ⊂ {1, 2, …}} of components of X. This will not be done in this paper though.

Let us recall that for two given matrices, say \(A = [a_{x_{k}x_{l}}]_{x_{k},x_{l}\in E_{1}}\) and \(B = [b_{y_{m}y_{n}}]_{y_{m},y_{n}\in E_{2}}\) indexed by elements of some finite sets E ₁,E ₂, its Kronecker product is the matrix \(A \otimes B = [(a \otimes b)_{(x_{k},y_{m})(x_{l},y_{n})\in E_{1}\times E_{2}}]\) with entries defined by \((a \otimes b)_{(x_{k},y_{m})(x_{l},y_{n})} = a_{x_{k}x_{l}}b_{y_{m}y_{n}}\). See, e.g., Horn and Johnson [8].

Ball, F., Milne, R.K., Yeo, G.F.: Continuous-time Markov chains in a random environment, with applications to ion channel modelling. Adv. in Appl. Probab. 26(4), 919–946 (1994). DOI 10.2307/1427898. URL http://dx.doi.org/10.2307/1427898 MathSciNetCrossRefMATH

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Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae. Electron. J. Probab. 18, no. 45, 21 (2013). DOI 10.1214/EJP.v18-2238. URL http://dx.doi.org/10.1214/EJP.v18-2238

Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Conditional Markov chains, part I: construction and properties (2015). URL http://arxiv.org/abs/1501.05531

Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Conditional Markov chains, part II: consistency and copulae (2015). URL http://arxiv.org/abs/1501.05535

Dabrowski, A.R., McDonald, D.: Statistical analysis of multiple ion channel data. Ann. Statist. 20(3), 1180–1202 (1992). DOI 10.1214/aos/1176348765. URL http://dx.doi.org/10.1214/aos/1176348765 MathSciNetCrossRefMATH

Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994). Corrected reprint of the 1991 original

Jakubowski, J., Niewęgłowski, M.: A class of \(\mathbb{F}\)-doubly stochastic Markov chains. Electron. J. Probab. 15, no. 56, 1743–1771 (2010). DOI 10.1214/EJP.v15-815. URL http://dx.doi.org/10.1214/EJP.v15-815 MathSciNetCrossRefMATH

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Jakubowski, J., Pytel, A.: The Markov consistency of Archimedean survival processes. J. Appl. Probab. 53(2), 293–409 (2015)MathSciNetMATH

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Title: A Note on Independence Copula for Conditional Markov Chains
Authors: Tomasz R. Bielecki
Jacek Jakubowski
Mariusz Niewęgłowski
Publisher: Springer New York
Book: Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science
Print ISBN: 978-1-4939-6968-5

Electronic ISBN: 978-1-4939-6969-2

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-1-4939-6969-2_10

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