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Extra info for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
Xn ≤ tn ) = FX (t1 , . . , tn ) = t1 tn ... −∞ −∞ fX (x1 , . . , xn )dx1 . . dxn . The formula can be interpreted as follows. The joint probability that the first random variable realizes a value less than or equal to t1 and the second less than or equal to t2 and so on is given by the cumulative distribution function F. The value can be obtained by calculating the volume under the density function f . Because there are n random variables, we have now n arguments for both functions: the density function and the cumulative distribution function.
1995). , New York: John Wiley & Sons. Bradley, B. and M. S. Taqqu (2003). ‘‘Financial risk and heavy tails,’’ in Handbook of Heavy-Tailed Distributions in Finance, S. T. Rachev, ed. Elsevier, Amsterdam, 35–103. , C. Kluppelberg and T. Mikosch (1997). Modeling extremal events for insurance and finance, Springer. , and G. Puccetti (2006). ‘‘Bounds for functions of dependent risks,’’ Finance and Stochastics 10(3): 341–352. , S. -W. Ng (1994). Symmetric multivariate and related distributions, New York: Marcel Dekker.
4) H= . .. .. . . . 2 ∂ 2 f (x) ∂ 2 f (x) . . ∂ ∂xf (x) 2 ∂xn ∂x1 ∂xn ∂x2 n 40 ADVANCED STOCHASTIC MODELS which is called the Hessian matrix or just the Hessian. The Hessian is a symmetric matrix because the order of differentiation is insignificant, ∂ 2 f (x) ∂ 2 f (x) = . ∂xi ∂xj ∂xj ∂xi The additional condition is known as the second-order condition. We will not provide the second-order condition for functions of n-dimensional arguments because it is rather technical and goes beyond the scope of the book.
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