Copulas and Markov processes

In this paper we study Markov processes using the copulas of A. Sklar [1], [2].A 2-copula is a function C: [0, 1] 2 --[0, 1] satisfying:(i) (Boundary conditions) and C(0, x2) C(Xl, 0) 0 for all Xl, x 2 [0, 1] C(Xl, 1) x and C(1, X2) X 2 for all Xl, X 2 [0, 1].(ii) (Monotonicity condition) C(Xl, x2) + C( y,, Y2) C(Xl, Y2) C( Yl, x2) > 0 for all Xl, x2, Yl, Y2 [0, 1] satisfying x _< Yl and x 2 < Y2"These conditions imply the continuity of C. Copulas are of interest because they link joint distributions to one-dimensional marginal distributions.Sklar showed that for any real valued random variables X and X 2 with joint distribution F12 there is a copula C such that (1.1) F12(Xl, X2) C(FI(Xl),F2(x2) )where F and F 2 denote the cumulative distribution functions of X and X 2 respectively.The copula C is said to connect XI and X 2 or to be a copula of X and X 2. In the other direction, for any distribution functions F and F 2 and any copula C, the function defined on the right hand side of (1.1) is a two dimensional distribution whose margins are F and F 2. Copulas thus capture all of the information concerning the dependence structure of random variables irrespective of their distributions and so provide a natural frame-