STOCHASTIC

A UNIFIED APPROACH TO STOCHASTIC DOMINANCE
we have g„ / g. Each g„ is a positive linear combination of functions of the
form^.^j.
Define/,(x) — j£ gn{y) dy. Since each g„ is a positive linear combination of
functions of the form /[.gw], each /„ is a positive linear combination of
functions of the form
Note that
and
Let
7 CySw]^ = (XAW)-W~.
f„ + (x) s j*g(y) dy = f + (x) for x ^ 0.
\g{o) • x for x g 0
K{X) U + W for x^O
; / N [^(°)' x for x ^ 0,
* = {/+(*) for x^O.
By the monotone convergence theorem, n(h„) / ^(/i) and v(h„) / v(/z). Since
v(//„) ^ /*(/;„) for each «, it follows that v(h) ^ n{h) ^ n(f + ), which is < oo
by hypothesis. Similarly, f„~(x)\ $$g(y) dy =f~(x) for x ^ 0. The function
e„(x) =yj, _ (jf)— g(o)x~ is a positive linear combination of functions of the
form x AW. Let e(x) =f~(x)—g(0)x~.
By the monotone convergence theorem, since 0 5; e„ \ e it follows that
v(e„) \ v(e) ^ v(/ _ ) and fi(e„) \ fi(e) ^ v(e).
Since/(x) = e(x) + A(x), it follows that v(/) ^ /*(/).
To prove the converse, let g(x) = x A W. Then n(g + ) < oo and # is concave
increasing. Consequently, by the hypothesis it follows that v(g) ^ n(g).
EXAMPLE 3
This example is called third-degree stochastic dominance, and was introduced
by Whitmore [14]. It shows that an investor with a concave increasing
utility function / with a nonnegative third derivative (provided / is thrice
differentiable) prefers X to Y if E\iX-w) 2 IiX&vl^ ^ E[(Y-w) 2 IlY^w{] for
each w.
1. STOCHASTIC DOMINANCE 105