STOCHASTIC

JAMES A. OHLSON
for all y,teSx'3C{Q,t0). Also, note that (S 2 + 2dt)(t~ 1 -l) > 0 for all
t e (0, t0). Therefore, it suffices now to show that
?- 3/2 exp{-i(5 2 + 250(?" 1 -l)} = exp{-|log/-i(« 2 + 2a/)(/- 1 -l)}-*0
as / -> 0, or, alternatively,
-^5 2 + 2dt)(t- 1 -l)
hm — = — oo.
,-o -flog?
That the last statement is true is easily shown by an application of L'H6pital's
rule.
The result just obtained can be extended to the general case where
F,(W) s $p,(s) ds is not a log-normal distribution function; i.e'. 0 < A, < 1
for at least one /'. The calculations to show this are lengthy, although straightforward.
Hence, only the argument is developed here; the formal verification
of the omitted step requires no more than ordinary differentiation, and the
specification of t0 is similar to that of the previous theorem. Assume, without
any loss of generality, that A£ > 0 for all i, and let pm(Xlt •••,Xm) denote the
density of the vector (Xu...,Xm)', where £C(X)~MLm(tfi,tX), and let
max A; = Aj. The density of p,{W) can be constructed from the (closed form)
density ptm{-) and a change of variables:
W=l4liXu
AT^Ar 1
Ml/*)] >
Y2 = 2,2 X2, or, equivalently X2 = Aj 1 y2j
v — l y Y — }~ x v
0 g \J\Pt{W)
= f P.M^LW-Y, yj.Vr,,...,, ^YJdY2- dYm,
JR'(W)
where |y| is the determinant of the Jacobian, which depends solely on A,
and R'(W) is the region of integration, which depends on W. Now if
W e [0, min Af) = St = [0, e), it can then be shown that there exists a
t0 = ?0 (ji, X) suc h that
log(1/0 + log/>,„(•) S log(l/f0) + logAom(-)
for a\\WeSuY2,...,Yme R'{W), and all 0 < / < t0. Thus, for any XeD,
230 PART III STATIC PORTFOLIO SELECTION MODELS