Dependence Between Components of Multivariate Markov Chains

Jacek Jakubowski^*

Modeling of evolution of dependence between processes occurring in some phenomena, e.g. financial markets, is important. Typically, one can identify marginal statistical properties of individual processes, and then one is confronted with the task of modeling dependence between these individual processes so that the marginal properties are obeyed. In my talk I present this modeling problem via the theory of Markov consistency and Markov copulae. So, I shall examine the problem of existence and construction of a multivariate Markov chain with components that are Markov chains with given laws. I give 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. I introduce and discuss the concept of weak Markov copulae. I examine relationship between the concepts of weak Markov consistency and weak Markov copulae, and the corresponding strong versions of these concepts. Weak and strong Markovian copulae provide, in a sense, dynamic counterparts of the classical concept of copulae in probability. I give an example of finite Markov chains that satisfy strong Markovian consistency, an example of a chain that satisfies weak Markovian consistency but does not satisfy strong Markovian consistency, and finally - an example of a chain that does not satisfy the weak Markovian consistency.\newline This talk will be based on the recent paper by T. Bielecki, J. Jakubowski and M. Niewęgłowski, Intricacies of Dependence between Components of Multivariate Markov Chains: Weak Markov Consistency and Weak Markov Copulae, to appear in Electron. J. Probab.

Mathematics Subject Classification: 60J27 60G55

Keywords: multivariate Markov chains; Markov consistency; Markov copulae; compensator of random measure

Minisymposion: Stochastic Models, Control and Applications