By Hult H., Lindskog F.

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If α is increasing and β is decreasing, then X1 and X2 are said to be countermonotonic. s. with T = F2← ◦ (1 − F1 ). 4 Covariance and linear correlation Let X = (X1 , . . , Xd )T be a random (column) vector with E(Xk2 ) < ∞ for every k. The mean vector of X is µ = E(X) and the covariance matrix of X is Cov(X) = E[(X − µ)(X − µ)T ]. Here Cov(X) is a d × d matrix whose (i, j)entry (ith row, jth column) is Cov(Xi , Xj ). e. the variance of Xi . 00 gbp Figure 15: log returns of foreign exchange rates quotes against the US dollar.

Dud −∞ and then f is called the density of F . The components of X are independent if and only if d F (x) = Fi (xi ) i=1 or equivalently if and only if the joint density f (if the density exists) satisfies d f (x) = fi (xi ). i=1 Recall that the distribution of a random vector X is completely determined by its characteristic function given by φX (t) = E(exp{i tT X}), t ∈ Rd . 1 The multivariate normal distribution with mean µ and covariance matrix Σ has the density (with |Σ| being the absolute value of the determinant of Σ) f (x) = 1 (2π)d |Σ| exp 1 − (x − µ)T Σ−1 (x − µ) , 2 Its characteristic function is given by 1 φX (t) = exp i tT µ − tT Σt , 2 43 t ∈ Rd .

First we simulate independently n times from the uniform distribution on the unit sphere to obtain s1 , . . , sn (above). Then, we simulate r1 , . . , rn from the distribution of R. Finally we put xk = rk sk for k = 1, . . , n (below). It follows immediately from the definition that elliptical distributed random vectors have the following stochastic representation. X ∼ Ed (µ, Σ, ψ) if and only if d there exist S, R, and A such that X = µ+RAS with S uniformly distributed on the unit sphere, R ≥ 0 a random variable independent of S, A ∈ Rd×k a matrix with AAT = Σ and µ ∈ Rd .

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Mathematical Modeling and Statistical Methods for Risk Management by Hult H., Lindskog F.

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