By Rodney Lester

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The epigraph eF in X x [0,1] of a distribution function F E D(X) is defined as follows. eF = cl ((x,p) E X x [O,lJ : p ~ F(x)} for FE D(X), where cl A denotes the closure of the set A. Consider first the expected value of a distribution F E D(X) given by E(F) = foM xdF(x). The following integration-by-parts argument shows that E(F) is the area of eF with respect to the Lebesgue measure. {M 10 xdF(x) = M (M xF(x)lo - 10 F(x)dx foM(l - F(x))dx dpdx. {M (l 10 1F(z) Similarily, the expected utility of F can be transformed to show that the expected utility representation equals the area of eF with respect to a product measure where the measure of an interval [x, yJ on the prize axis is given by u(y) - u( x) and the measure on the probability axis is again the Lebesgue measure.

7) to be a generalized utility function are as follows. Firstly, by property l(i) v must vanish on the set B, hence u(O) = o. By first-order stochastic dominance u is strictly increasing and 9 > o. By property l(ii) the derivative VI must be strictly increasing in the propbability p. The derivative VI is given by v (z ) = { (u'(z) + u(z)g'(z)lnp)p9(z) if p =I 0 1 ,p 0 ·f P -- 0 . 1 50 Differentiation3 with respect to the second variable yields V12(X,P) = ((ug)'(x) The function V12 + u(x)g(x)g'(x)lnp)p9{:Z:)-l.

M (l 10 1F(z) Similarily, the expected utility of F can be transformed to show that the expected utility representation equals the area of eF with respect to a product measure where the measure of an interval [x, yJ on the prize axis is given by u(y) - u( x) and the measure on the probability axis is again the Lebesgue measure. 14) corresponds to the case where the measure JL is a general product measure. Integrating by parts the anticipated utility of F and assuming u(O) = 0 yields 1M 10 u(x)d(h 0 F)(x) = M (M u(x)(hoF)(x)lo - 10 (hoF)(x)du(x) foM(l - (h {M (l 10 1F(z) 34 0 F)(x))du(x) dh(p )du( x).

### Introduction to the Insurance Industry by Rodney Lester

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