STOCHASTIC

THE ASYMPTOTIC VALIDITY OF QUADRATIC UTILITY
III. The Log-Normal Case
In this section the "compactness" characteristic of the log-normal distribution
is derived. First the dominated convergence of the continuous density
(\jt)pt{W) as t, W->Q is established; subsequently, exact order properties of
E{W— 1)" for n = 1,2,..., are proved. It is assumed that (logX,, ....logA^)'
is distributed in the multivariate normal form with parameters
and
(ElogXi,...,ElogXm)' s {tnu...,tnm) = tii
[cov(log*i, log*})] = [/*,] = /£.
In other words, the expected growth and variance of growth are linear in time,
and all increments are independently distributed. For obvious reasons, this
process is referred to as a stationary "geometric" Wiener process (see Merton
[4]). As it turns out, the Xt's, and W =^dXiXi, are not quite "compact" in
the sense of Samuelson; i.e., assumption Al is not satisfied. However, the
moments are sufficiently well behaved to assure quadratic utility as / -> 0;
i.e., they will satisfy Theorem 1 and Corollary II for all A e D. For simplicity,
we first analyze the univariate case and write ££{X) ~ MLl (///, ta 2 ) to denote
that the probability law of X is log-normal with parameters tfi and ta 2 .
Theorem III Let p,(x) and p0(x) be the densities of two (independent) lognormal
random variables specified by MLi{tfi,ta 2 ) and ML1(0,1), respectively.
Then, there exist a t0 and e such that
for all t e (0, /0) and x e [0, e) = Su
(llt)Pt(x)^p0(x)
Proof The density pt(x) oc x~ l {ta 2 y 1 ' 2 exp{-i(logx-/^) 2 (m 2 ) -1 };
hence if the function g(y, t) is defined by
9{y,t) = Pt{x){ta 2 YI 2 lp0{x)Y x = exp{-i[(^-/,i) 2 (rff 2 )- 1 -/]}
where y = log x, then it suffices to show that
t ~ 1 v (x)
0 yeSi' Po( X )
where Sx' = (—oo,loga), t e (0, t0), and g(y,t) is defined on 5,'^(0,/0).
Let t0 = inf{|, |^|,|l//i|,o-" 2 } and let loge = S be any fixed number less
than — 1. It is then straightforward to verify that
sup g(y,t) ^ cxp{-i(5 2 + 2dt)(r l -l)}
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 229