Stochastic Programming - Index of

50 STOCHASTIC PROGRAMMING
Figure 20 P here is not quasi-concave: P (C) = P ( 1 A + 2 B) = 0, but
3 3
P (A) =P (B) = 1 . 2
in IR 1 every monotone function is easily seen to be quasi-concave, such that
every distribution function of a random variable (always being monotonically
increasing) is quasi-concave.But not every probability measure P on IR is
quasi-concave (see Figure 20 for a counterexample).
Hence we stay with the question of when a probability measure—or its
distribution function—is quasi-concave. This question was answered first for
the subclass of log-concave probability measures, i.e. measures satisfying
P (λS 1 +(1− λ)S 2 ) ≥ P λ (S 1 ) P 1−λ (S 2 )
for all convex S i ∈F and λ ∈ [0, 1]. That the class of log-concave measures is
really a subclass of the class of quasi-concave measures is easily seen.
Lemma 1.2 If P is a log-concave measure on F then P is quasi-concave.
Proof Let S i ∈F, i =1, 2, be convex sets such that P (S i ) > 0, i =1, 2
(otherwise there is nothing to prove, since P (S) ≥ 0 ∀S ∈F). By assumption,
for any λ ∈ (0, 1) we have
P (λS 1 +(1− λ)S 2 ) ≥ P λ (S 1 ) P 1−λ (S 2 ).
By the monotonicity of the logarithm, it follows that
ln[P (λS 1 +(1− λ)S 2 )] ≥ λ ln[P (S 1 )] + (1 − λ)ln[P (S 2 )]
≥ min{ln[P (S 1 )], ln[P (S 2 )]},