STOCHASTIC

THE ASYMPTOTIC VALIDITY OF QUADRATIC UTILITY
That is, if A**(/) is the solution to the surrogate problem
max{EU(W)-EQ3}, (4)
XeD
then ||A**(0-^*(OII -*• 0 as t -»• 0. One set of assumptions about F and U is
implied in the cited Samuelson paper [7]. The restrictions on Fare given by
certain compactness characteristics:
A\:
E(W-1) _ A E(W-1)"
1^F^ ;:;' EOV- » = -B, lim^^r= t^Cn; n = 3,4,...,
2 ~B ; , £(^-i) 2
(5)
where C„ P 0 remains bounded as ? -> 0. 2
Alternatively, if O(-) denotes the usual asymptotic order symbol meaning
"the same order as," and o(-) denotes "smaller order than," then assumption
Al entails C„ = 0(1), E{W-\) n = 0{t), n = 1,2, and E(W-l)" = 0(t" /2 )
[ = o(0],« = 3,4,.... 3
As hypotheses on U{W), it is first assumed that f/is endowed with an exact
Taylor's infinite expansion, i.e.,
A2:
EU(W) = E f U\\) (W- \yjj\. (6)
Further, it is assumed that the expectation operator can be interchanged with
the summation operator:
A3:
CO CO
EU(W) = £ V^\\)E(W-\yiJ\ = £ hj(t). (7)
j=0 j=0
As a final requirement, if it is assumed (or proved) that
AA:
limlim £ A;(0 = lim £ hj(t), (8)
f-»0 n->oo j = 3 l/ttn->cc j—3
then the convergence is uniform and ~£f=3hj(t) = o(t). It is now obviously
true that assumptions A1-A4 imply that (3) holds since
EU(W) = t/(l) + U (i \\)E(W-1) + \U (2 \\)E{W- \f + o(t), (9)
and the first two moments are of order O(t).
2 See Samuelson [7]. Note that Samuelson's symbol "a" corresponds to *Jt in this paper.
3 Define 0(t") = h(t) ifflim^o I' ~"*(0I = K> 0, and h(t) = o(l) iff ]im,-.0 \t~ '/KOI = 0.
Further, note that h(t) = 0(t") implies h(t) = o(t) iff n > 1.
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 223