STOCHASTIC

SEPARATION IN PORTFOLIO ANALYSIS
Theorem 1 A necessary condition for local or global separation of problem
Pk is that marginal utility satisfy either
or
for all £ e Ek, with
Au'iO" + Bu'{& = £ (with a * P)
(0, oo) if k = n.
For general discrete random returns (u, oo) with if N k > < n, n, there are two cases which
must be considered: (1) markets without money, that is, markets having no
risk-free asset, and (2) markets with money. Each case is treated separately.
Case 1 To derive necessary conditions for separation in general markets
without money, it is enough to derive these conditions for a special subclass
of such markets. In particular, suppose that the matrix p has the structure [2]:
[P] =
0-0
0 0
0 0
Pn Pn 0 is arbitary. Assuming an interior solution (which can always be obtained
by suitable choice of p) it follows as in (19) that
"- 1 n-^i \
7=1 \Pjk=l '
where R = p i . There are two subcases:
(i) g(z) = Az" + Bz p . Equation (29) becomes
where
J = l
(27)
(28)
3. SEPARATION THEOREMS 163
(29)
(30)
(31)