Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae

In this paper we examine the problem of existence and construction of multivariate Markov chains such that their components are Markov chains with given laws. Specifically, we provide 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. Accordingly, we introduce and discuss the concept of weak Markov copulae. Finally, we examine relationship between the concepts of weak Markov consistency and weak Markov copulae, and the corresponding strong versions of these concepts.

[1]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[2]  R. Nelsen.  An Introduction to Copulas , 1998 .

[3]  R. Ryan.  Introduction to Tensor Products of Banach Spaces , 2002 .

[4]  Abe Sklar,et al.  Random variables, joint distribution functions, and copulas , 1973, Kybernetika.

[5]  T. Bielecki,et al.  A Markov copulae approach to pricing and hedging of credit index derivatives and ratings triggered step-up bonds , 2008 .

[6]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[7]  A. Cousin,et al.  Dynamic Modeling of Portfolio Credit Risk with Common Shocks , 2011 .

[8]  L. Rogers,et al.  Diffusions, Markov Processes and Martingales: References , 2000 .

[9]  T. Bielecki,et al.  Study of dependence for some stochastic processes: Symbolic Markov copulae , 2012 .

[10]  Agatha Murgoci.  Essays in mathematical finance , 2009 .

[11]  Mu-Fa Chen,et al.  From Markov Chains to Non-Equilibrium Particle Systems , 1992 .

[12]  Frank Ball,et al.  Lumpability and marginalisability for continuous-time Markov chains , 1993 .

[13]  Jacek Jakubowski,et al.  Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes , 2010 .

[14]  E. B Dynkin,et al.  Markov Processes: Volume 1 , 2012 .

[15]  M. Rosenblatt,et al.  A MARKOVIAN FUNCTION OF A MARKOV CHAIN , 1958 .

[16]  M. Yor,et al.  Mathematical Methods for Financial Markets , 2009 .

[17]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[18]  Jacek Jakubowski,et al.  Study of Dependence for Some Stochastic Processes , 2008 .