By Charles Castaing, Marc Lavie (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

A lot of monetary difficulties should be formulated as limited optimizations and equilibration in their strategies. a number of mathematical theories were delivering economists with vital machineries for those difficulties coming up in monetary concept. Conversely, mathematicians were prompted via numerous mathematical problems raised by means of monetary theories. The sequence is designed to compile these mathematicians who're heavily attracted to getting new not easy stimuli from financial theories with these economists who're looking powerful mathematical instruments for his or her research.

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On A, by integrating on A we deduce that δ ∗ (x, co w-ls A Sn (ω) )dP (ω) ≥ n A δ ∗ (x, Z∞ (ω))dP (ω) that contradicts the inequality δ ∗ (x, co w-ls A Sn (ω) )dP (ω) < n A δ ∗ (x, Z∞ (ω))dP (ω). Remarks. 2 hold true when E is separable B convex reflexive Banach space. 2 needs a special treatment. 6. For shortness we omit the details. 1 involving the support function of an epigraph multifunction and the support function of the conditional expectation of this epigraph. It seems difficult to obtain the above convergence results in the case when the random sets are unbounded.

T, s 0 Then f˜(t, s) is continuous in (t, s). Let 0 t f˜(t, s1 ) − f˜(t, s2 ) = exp( s1 < s2 . Then ˜ s1 ) − M(t, ˜ s2 )), ξ(r)dr)(M(t, t ∈ [0, s1 ]. s. ˜ s1 ) − M(t, ˜ s2 ) = 0} ∧ s1 . s. s. t ∈[0,s1 ] for any s1 , s2 ∈ Q with s2 > s1 0. So there is an 1 ∈ F with P ( 1 ) = 1 such that f˜(t, s1 , ω) > f˜(t, s2 , ω) for any ω ∈ 1 , s1 , s2 ∈ Q with s2 > s1 > 0 and t ∈ [0, s1 ). Since f˜(t, s) is continuous in (t, s), we see that f˜(t, s) is non-increasing in s. So we see that f˜(t, s1 , ω) > f˜(t, s2 , ω) for any ω ∈ 1 , t, s1 , s2 ∈ [0, ∞) with s2 > s1 > t.

Since F (y, 0, u) = y − u, Dy u(0, ˆ 0) is regular, and thus, D u(0, ¯ 0) is also regular. Hence, by the inverse function theorem, there exist open U5 ⊂ U4 and U6 ⊂ Rn such that (0, 0) ∈ 10 Note that ε can be taken uniformly on U whenever U is sufficiently small. 1 1 1 48 Y. Hosoya U5 ∩ U6 , u¯ is a C q -class bijection from U5 onto U6 , and u¯ −1 is also C q -class. Without loss of generality, we assume that U6 ⊂ U4 and U6 = V6 × W6 for some open V6 ⊂ Rm and W6 ⊂ Rn−m . Fix any y and choose any (t, z) such that (y, z) ∈ U4 and t ∈] − 2, 2[.

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