An Introduction To Copula And Its Application

The dependence between random variables is completely described by theirjoint distribution.If multivariate distribution could be precisely constructed, anyresearches on random variables would become comparative easy. When eachmarginal distribution comes from the identical standard distribution, such asNormal or Student-t distribution, we can construct the joint distribution with thecorrelation information. While the hypothesis of each marginal distribution com-ing from identical distribution often fails, for example, X1～N(μ,σ2),X2～t(ν),in addition to the correlation coe?cientρ, how should the joint distribution beconstructed? Even each marginal distribution comes from identical distribution,but not standard distribution, how to deal with this situation?Consistency and dependence measurement are always taking important rolesin many fields. If we know that the correlation coe?cient between X1 and X2isρ, then Y1 =α1(X1) and Y2 =α2(X2),where functionαhas arbitrary form,is the dependence between Y1 and Y2 identical to X1 and X2 ? Compared withlinear correlation, what's the advantage of rank correlation coe?cient? When weperform Monte Carlo simulation, how to generate arbitrary marginal distributionunder specific correlation coe?cient?According to the above questions,Copula theory provides good solutions.Copulae are functions that describe dependence among variables, and provide away to create multivariate distributions by specifying marginal univariate distri-butions and choosing a particular copula family to model correlated multivariatedata. The copula of a multivariate distribution can be considered to be the partdescribing the dependence structure. Furthermore, strictly increasing transfor-mations of the underlying random variables result in the transformed variableshaving the same copula. Hence copulae are invariant under strictly increasingtransformations of the margins. This provides a way of studying scale-invariantmeasures of associations and also a starting point for construction of multivariatedistributions.This paper systematically summarizes copula theory including its history,its mathematical traits, its main classification, its construction methods, its pa- rameters estimation theory, and how to apply copula theory to generate desiredrandom values.Based on deeply insight into copula theory, this paper focuses creativity onthe following aspects:1, Systematically summarizing the history of copula theory, and its applica-tion in security analysis, credit risk analysis, system stability analysis, biostatis-tics, etc.2, Based on deep study on correlation theory, it analyzes and proves the rela-tionship between copulae and often used correlation index,and presents copula'sadvantages for modelling dependence.3, Providing algorithms to calculate rank correlation coe?cient with theknowledge of linear correlation coe?cient,and vice versa.4, Combining copula theory with asymmetric GARCH Model, Monte Carlosimulation Model, General Pareto Distribution and EVT theory to simulatejointly distribution of Shanghai Stock return index and Shenzheng Stock re-turn Index.