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**Extra resources for Stochastic Optimization Models in Finance**

**Example text**

7. (Jensen's inequality) Suppose Λ <= E" and H c Em are convex sets. (a) Show that the function f(x) : Λ -» ( — oo,+oo]is convex on Ω if and only if Σ *i/(*') ^ f\ Σ hA, i=l \f=l where A, £ 0, Σ ^ = 1. /=1 / (b) Show that the function /(£) ://->(— oo,+oo] is convex on H if and only if provided ξ exists. (c) Show that the inequalities are reversed if/is concave. (d) Show that these inequalities become equalities if/is linear. (e) Show that the inequality in (b) becomes an equality for all / i f and only if all of the mass of ξ is at ξ.

Two notions are present in this definition. First, the dependence on δ has been suppressed; only νδ remains. This is justified by the lemma, which assures that if δ and y are two policies such that νδ = νγ9 then νπ(χ) = νλ(χ) where π and λ are defined by πχ = λχ = dx and, for all z Φ x, nz = δζ and λζ = yz. Second, the dependence on dx in h(x,dx,vô) has been made explicit. Note that h(x, dx,vô) is not a function of δχ—that decision δχ is immaterial, and it is not required that δχ = dx. In economic terms, h(x,dx,vô) might be inter preted as the cumulative return obtained by starting at state x and choosing decision dx with the prospect of receiving the terminating reward νδ(ζ) if transition occurs to state z.

Let* 1 , x 2 e T . Then V f l ^ X x 2 - * 1 ) = (V^(/(x1),^(x1))V/(x1) + + V^C/Xx 1 ), ^(x 1 ))V^(x 1 ))(x 2 -x 1 ) (by the chain rule) < V ^ t / X x 1 ) , g(xl))(f(x2) -/(x1)) + + V 2 (p(/(x 1 ), â f(x 1 ))fe(x 2 )-^(x 1 )) (by the convexity of / , concavity of g, increasingdecreasing property of φ and the Lemma). Hence V ö ( x 1 ) ( x 2 - x 1 ) > 0 => V 1 ç>(/(x 1 ),^(x 1 ))(/(x 2 )-/(x 1 )) + + V ^ i / i x 1 ) , g(xx)){g(x2) - g{x1)) > 0 (by the above inequality) =>

Φϋϊ*1), *(*')) (by the pseudo-convexity of φ) => 0(x 2 ) > OCX1) , 117 PART I.