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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- 1.
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.
- 2.
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].
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Acknowledgements
We thank the referees and the editors for valuable comments and suggestions, which we used revising the original version of this note.
Research of T.R. Bielecki was partially supported by NSF grant DMS-1211256.
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Bielecki, T.R., Jakubowski, J., Niewęgłowski, M. (2017). A Note on Independence Copula for Conditional Markov Chains. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_10
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