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Extra info for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)
Un ) . ∂u1 . . ∂un In this way, using the relationship between the copula and the distribution function, the density of the copula can be expressed by means of the density of the random variable. This is done by applying the chain rule of differentiation, c(FY1 (y1 ), . . , FYn (yn )) = fY (y1 , . . , yn ) . fY1 (y1 ) . . 4) In this formula, the numerator contains the density of the random variable Y and on the denominator we find the density of the Y but under the assumption that components of Y are independent random variables.
Un ) = u1 . . un . This copula characterizes stochastic independence. Now let us consider a density c of some copula C. 4) is a ratio of two positive quantities because the density function can only take nonnegative values. For each value of the vector of arguments y = (y1 , . . 4) provides information about the degree of dependence between the events that simultaneously Y i is in a small neighborhood of yi for i = 1, 2, . . , n. That is, the copula density provides information about the local structure of the dependence.
In the two-dimensional case only, the lower Fr´echet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables. In a similar way, the upper Fr´echet bound, sometimes referred to as the maximal copula, represents perfect positive dependence between the two random variables. 8 SUMMARY We considered a number of concepts from probability theory that will be used in later chapters in this book. We discussed the notions of a random variable and a random vector.