Gaussian Copula Processes And Its Applications

The present work concerns the study of the constructor methods for Copula process and the related applications for Copula process. Copula process is mainly used to study the dependency of stochastic process. Copula process can connect random variables which has any marginal distribution with a specific related structure. Copula process is an extension of the copulas in the field of stochastic process.On the basis of discussing Copula process, we further studied the Gaussian copula process, which is a stochastic process consist of copulas with Gaussian form. As an example, we develop a stochastic model, Gaussian Copula Process Volatility(GCPV), to predict the latent standard deviations of a sequence of random variables.The principle of GCPV is as follows: Assume volatility distributed as a Gaussian copula, through an inverse warping function to mapping the standard deviation to a latent space and make it better fitted a Gaussian process. Then using Laplace approximation to approximate the Gaussian process, subsequently to predict volatility. So we can separate the dependency structure between volatilities from their marginal distributions.Volatility Models have a very wide range of financial applications. GARCH model is a very important model in the study of volatility, and it could effectively fit the heteroscedasticity for long-term memory. In this article, we compare GARCH model and GCPV model through simulation experiment. The results show that the GCPV model outperforms the GARCH model, especially for the jump points, the relation between volatility implied by GCPV model of is more accurate than GARCH model. GCPV model also has the following advantages: We can effortlessly handle missing data; we can easily incorporate covariates other than time(like interest rates) in our covariance function, which is widely applied.