By Shigeo Kusuoka, Toru Maruyama

ISBN-10: 4431541136

ISBN-13: 9784431541134

ISBN-10: 4431541144

ISBN-13: 9784431541141

The sequence is designed to assemble these mathematicians who're heavily drawn to getting new hard stimuli from fiscal theories with these economists who're looking potent mathematical instruments for his or her examine. loads of financial difficulties should be formulated as limited optimizations and equilibration in their recommendations. quite a few mathematical theories were providing economists with vital machineries for those difficulties coming up in monetary conception. Conversely, mathematicians were influenced via a variety of mathematical problems raised by way of financial theories.

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The Annals of Probability 31, 63–92 (2003) 18. Dudley, R. : Real Analysis And Probability, Chapman-Hall, Mathematics Series. Wadsworth, Inc. (1989) 19. : Weak-star convergence of convex sets. Journal of Convex Analysis 13(3 + 4), 711–719 (2006) 20. : Integrals, conditional expectations and martingales of multivalued functions. J. Multi. Anal. 7, 149–182 (1977) 38 C. Castaing and M. Lavie 21. Garling, D. J. : Subsequence principles for vector-valued random variables. Math. Proc. Cambridge Philos.

S↑∞ Proof. Note that for 0 s1 s2 E[ sup |M(t, s1 ) − M(t, s2 )|4 ] t ∈[0,∞) s2 4E[| s1 ξ(r)dr|4 ] 4E[|M(s1 , s1 ) − M(s2 , s2 )|4 ] 4(s2 − s1 )3 ( s2 s1 E[ξ(r)4 ]dr). 58 S. Kusuoka and T. Nakashima So by Kolmogorov’s continuity theorem and the assumption (B-1), we see ˜ ∗, ω) → that there is a M˜ : [0, ∞) × [0, ∞) × → [0, ∞) such that M(·, ˜ s) = M(t, s)) = 1, t, s 0. Let [0, ∞) is continuous and P (M(t, t ∧s f˜(t, s) = exp( ˜ s) ξ(r)dr)M(t, 0. t, s 0 Then f˜(t, s) is continuous in (t, s). Let 0 t f˜(t, s1 ) − f˜(t, s2 ) = exp( s1 < s2 .

S. 0 (2) For any ω ∈ f (·, ∗ : ω) : [0, ∞) × [0, ∞) → [0, ∞) is continuous. (3) If s2 > s1 > t, then f (t, s1 , ω) > f (t, s2 , ω) > 0, ω ∈ . (4) For any t 0 f (t, t) = 1, lim f (t, s) = 0. s↑∞ Proof. Note that for 0 s1 s2 E[ sup |M(t, s1 ) − M(t, s2 )|4 ] t ∈[0,∞) s2 4E[| s1 ξ(r)dr|4 ] 4E[|M(s1 , s1 ) − M(s2 , s2 )|4 ] 4(s2 − s1 )3 ( s2 s1 E[ξ(r)4 ]dr). 58 S. Kusuoka and T. Nakashima So by Kolmogorov’s continuity theorem and the assumption (B-1), we see ˜ ∗, ω) → that there is a M˜ : [0, ∞) × [0, ∞) × → [0, ∞) such that M(·, ˜ s) = M(t, s)) = 1, t, s 0.

### Advances in mathematical economics by Shigeo Kusuoka, Toru Maruyama

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