By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to assemble these mathematicians who're heavily drawn to getting new not easy stimuli from financial theories with these economists who're looking powerful mathematical instruments for his or her examine. loads of monetary difficulties might be formulated as limited optimizations and equilibration in their suggestions. numerous mathematical theories were providing economists with quintessential machineries for those difficulties coming up in financial thought. Conversely, mathematicians were prompted by means of a variety of mathematical problems raised by way of fiscal theories.

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Let N 3, M 1, and be a non-void open subset in RM . Then the subset of C(N) ( ) whose elements are admissible families of copula functions is a set of the first category in Baire’s sense. Credit risk models and copula 57 We also show that a family of Gumbel copula functions of 3 variables is not admissible by relying on numerical computation in Sect. 7. Acknowledgement The authors thank the referee for useful suggestions and comments. 2. Preliminary results Let ξ : [0, ∞) × → [0, ∞) be a G-progressively measurable satisfying the following three conditions.

The next corollary clearly holds. 5 See 2) of Theorem 1 in sect. 2. 6 In this note, x i denotes the i-th coordinate of vector x. 7 The term f being C 0 -class means that f is continuous. 42 Y. Hosoya Corollary 1. The condition (A-∞) at x, (B-∞) at x, and (C-∞) at x are mutually equivalent. We should mention that if q < ∞, then both conditions (B-q) and (C-q) are not the sufficient condition of (A-(q + 1)). Debreu [4] presents a counterexample. He constructs the following g1 : ⎧ ⎨( √ y 2 , √ 1 ) (if y ≥ 0) 1+y 4 1+y 4 g1 (x, y) = ⎩(0, 1) (otherwise) and claims that there is no C 2 -class real-valued function u such that Du(x, y) = λ(x, y)g1 (x, y) for any (x, y) around (0, 0).

Castaing and M. Lavie lim d(x, n→∞ 1 n n Xγ (i) (ω)) = d(x, X∞ (ω)) ∀x ∈ E. s. i=1 Proof. Step 1 Let (ek )k∈N be a dense sequence in B E . For each fixed k ∈ N and n ∈ N, pick an integrable selection uk,n of Xn such that ek , uk,n (ω) = δ ∗ (ek , Xn (ω)) ∀ω ∈ . s. s. i=1 with respect to the norm of E. s . s. 2 A more general version of Koml´os theorem for B convex reflexive Banach space is available in [21]. s. s. s. n→∞ ∀x ∈ E. Finally X∞ is measurable and satisfies the inequality |X∞ |dP ≤ lim inf |Yn |dP ≤ sup n n |Xn |dP < ∞.