1

Let (X1, X2) be a pair of standard exponential random variables with the countermonotonicity copula and let U be a standard uniform random variable. a) Find a stochastic representation of the form (X1, X2) = (f(U), g(U)) for functions f and g which shows how (X1, X2) can be simulated. b) With the help of a), use software and numerical integration to calculate the minimal correlation for two standard exponential random variables.Let : [0, co) - [0, 1] be a generator of a d-dimensional Archimedean copula C. a) Show that lime-+0 v(t) =0 must hold. b) Show that (0) = 1 must hold. c) Show that (ct), t 2 0, generates C for all c > 0. d) Show that the lower tail dependence coefficient of the bivariate copula satisfies A = (21)a Show that every coherent risk measure o : M - R is a convex risk measure, that is o satisfies o(AL1 + (1 - A)L2) 0. Hint. You may use the Lambert W function, defined as the inverse of the function r rexp(I). b) Compute ea(L) for L ~ Pa(2, k), k > 0