Copula-based Stationary Markov Modeling And Applications

In the traditional method, one study Markov process by giving the initial distribution of the random variable and the corresponding transition probabil-ity satisfying the Chapman-Kolmogorov equation. In this paper, we combine copula functions with Markov process organically and define products, which we call*operation on copulas and the corresponding generalized*product on copulas. We obtain the copula characterization of Markov process by giving a series of copulas satisfying certain conditions and the marginal distribution of the random variable. On the basis of the former study, we construct a class of copula-based semiparametric stationary Markov models and apply this model to the computation of conditional quantile, which is the core content of risk management in financial market. Because of the flexibility and effectiveness of copula modeling in stochastic process, we are no longer constrained by the assumption of normal marginal distribution. We make use of nonparametric kernel density estimation method to estimate the marginal distribution and specify the copula parametrically, and then we estimate the dependence pa-rameter α by canonical maximum likelihood method. In order to choose the most appropriate copula, we refer to a composite criterion, which combine A-IC criterion and the smallest error criterion, which is calibrated by the sum of absolute error between estimated copula and empirical copula. After this, we can use the corresponding dependence parameter estimation to estimate the conditional quantile. In the end, we simulate the asymptotic distribution of the copula dependence parameter using bootstrap method and study the large sample property of the estimations. Obviously, the most concerned index is the return and the risk of the investors’financial position. No matter which investment strategy to be taken, the target is to get more return at lower risk level. Therefore, the study of dependence between different financial positions is always very important. There are many a excellent papers and research results in this field. In this paper, we transform the multivariate time series problem into pseudo univariate time series problem, and then apply the former described copula-Markov semiparametric model to it. In the part of empirical research, we apply our model to the return series of Shanghai Composite Index and HongKong Hang Seng Index, and study their dependence structure.