STOCHASTIC

ON THE EXISTENCE OF OPTIMAL POLICIES UNDER UNCERTAINTY 37
forward "perfect market" assumption. Note that each gamble g' maps 5
into C, and the set of gambles G maps S into C" 6 R n , where C" is the
Cartesian product of the sets C, i = /,..., n.
An element x of the feasible action or feasible gambling position set X
is characterized by a vector (x 1 ,..., x") representing the number of units
of gambles (g 1 ,—, g") held. Each element of X, therefore, maps a point C",
depending upon the state of nature, into a set of outcomes C C R 1 , where
C = {c = G(s)' x | G(s) eC n ,xe X),
and G(s) is the column vector of net return functions g 1 (s),...,g"(s). As
in the general decision problem, the choice of an action x e X is a choice
of mapping from 5 into C, but via the gambling opportunity set G.
We assume actions can be ranked by expected utility. 2 That is, the
ranking is consistent with a utility function over C, and a subjective
probability measure over S, such that
xx > x2 iff E{U[G(s)' jr,]} 3= £{U[GW xj}.
We impose the following restrictions on the feasible action set X and
on the preference relation over X, as reflected by the utility function over C:
XA
X.2
X.3
properties
(7.1
(1)
0)
U.2
X is closed and (perhaps weakly) convex.
0 e X, where 0 is the origin.
There exists an H < oo such that there is no xk e X with the
G(s)' xk > 0, except on a set with probability measure zero,
and
|] xk || > //, where || xk || is the Euclidean norm of xk .
U'(c) > 0 for all c e C;
t/"(c) < 0 for all c e C.
Assumption AM necessarily limits the class of problems considered, but
we shall indicate several important economic problems which satisfy this
requirement. Typically, the feasible action set may be affected by prices
of gambles, initial wealth, technological relationships, etc. Assumption X.2
states that no gambling at all is a feasible action. Assumption X.3 states
that there is no arbitrarily large gambling position which offers a positive
or zero return with probability one. It is equivalent to Hakansson's
2 Axioms on a preference ranking leading to the conjoint imputation of a subjective
probability distribution over states of nature and a utility function over outcomes are
discussed by Savage [7], His analysis requires S to have an infinite number of elements.
2. EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES 269