# Credit risk: modeling, valuation and hedging by Tomasz R. Bielecki

By Tomasz R. Bielecki

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Example text

Td_l 1, cases.

E. P{X[ ~ m} ~ 1/2 and P{X[ ~ m} ~ 1/2. 6 There are positive constants elements of matrix A such that (aj if aii = 0 for all i = 1,2, ... 7) (bj if aii =I=- 0 for some i = 1,2, ... , n, then for any u ~ 0 we have P{X[ ~ u + C2 (n -1) m} ~ 2n - 1 P{IQxl ~ CI u}. 8) PROOF. Case a: Since A is not a zero matrix, there exists aij =I=- 0 with i =I=- j. Without lose of generality, we may assume al2 =I=- O. 3, we have for any positive u Case b: Without loss of generality, we assume all a = 4 CI (A) (n - =I=- O.

Further, without loss of generality we assume that the random vector (Xl, X2, .. ,Xd) has the standard uniform marginal distributions [Le. 7)]. 1, the function B(d-2) is convex. Therefore, B(d-2) is absolutely continuous and d-l p (2: R(Xi) E ~d-l) = o. i=l Introduce for all y E ~d-l and for all Xl, X2, ... s. the conditional distribution function of the sum 29 On Finite-Dimensional Archimedean Copulas d P(LR(Xi)