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Copulas and Markov processes. (English) Zbl 0770.60019
For any real-valued random variables \(X\) and \(Y\) with joint distribution \(H\), A. Sklar [Kybernetika, Praha 9, 449-460 (1973; Zbl 0292.60036)] showed that there is a cuopla \(C\) such that \(H(x,y)=C(F(x),G(y))\), where \(F\) and \(G\) are the marginal distributions of \(X\) and \(Y\) and \(H\) is their joint distribution. Consequently, copulas carry complete information on dependency relations between \(X\) and \(Y\). The authors define a new binary operation on the set of copulas, study its algebraic properties, and show how this operation can be used to study Markov processes. This approach to Markov processes is quite different from the standard approach. Instead of giving an initial distribution and the transition probabilities, all the marginal distributions and a family of copulas satisfying the authors’ conditions are specified.

60E99 Distribution theory
60J25 Continuous-time Markov processes on general state spaces
60J65 Brownian motion
60J05 Discrete-time Markov processes on general state spaces
60G07 General theory of stochastic processes