STOCHASTIC

ON THE EXISTENCE OF OPTIMAL POLICIES UNDER UNCERTAINTY 41
To show y(z) e Y, we must show Hy(z) e Z. As both 0 (the origin),
X(z) y(z) e Z, the convexity of Z implies Hy(z) e Z, because H e [0, A(z)].
THEOREM I. Lef G, X, and U satisfy G.l, G.2; AT.1, A^.2, Af.3; U.l, f/.2,
am/ £/.3A: £/(c) bounded above. Then, for every yeY, there exists a X*(y)
such that V[X*(y)y] = V(0), and V(Xy) < V(Q)for A > X*(y).
Proof. Let j9(s, y) = G(s)' y. Note for yeY, pr(fi < 0) > 0, by X.3.
First, we wish to show for any je Y, we may find a A **(>>) such that
V[X**(y)y] < V(0). From the definition of V, we have
From U.2,
V(Xy) - V(0) = Prtf < 0) £[C/(A]8) - £7(0) | ]8 < 0]
+ Prtf > 0) £[t/(Aj8) - 1/(0) | j8 > 0]. (1)
Pr(P < 0) E[U(\p) - t/(0) | ^ < 0] s£ PKjg < 0) £[A£C'(0) | j8 < 0]
= \U'(0) E\fi | j8 < 0] i>r(/9 < 0)
= XKX , say, where Kx < 0.
The second RHS term of (1), which is positive, clearly is less than
C — 1/(0) = K2, where C is the upper bound of U(c). Therefore, by
choosing X**(y) > -KJ^ (itself a function of y), V[X**(y)y] < K(0).
By the continuity of V in A, and the fact that V(Hy) > V(0) and
V[X**(y)y] < K(0), there exists a A*(j>) e [H, X**(y)] such that
K[A*(y)j]= K(0).
Now assume that for some A 0 > X*(y), V(X"y) ^ V(0). By Proposition II,
V(Xy) is strictly concave in A. Therefore V[X*(y) y] > ^(0), as
\*(y) e (0, A 0 ). But this is a contradiction, and we conclude for A > X*(y),
V(Xy) < V(0).
THEOREM II. Let Gsatisfy GA, G.2; andG.3: E[g'(s)] < oo,i= \,...,n.
Let X satisfy X. 1, X.2, and X-.3;
Let U satisfy U.\, U.2, and C/.3a: lim U'(c) = 0.
Then, as in Theorem I, there exists for every yeY a X*(y) such that
V[X*(y)y] = V(0), a"d V(Xy) < V(0)for A > X*(y).
Proof. As in Theorem I, (1) holds and the first RHS term will be less
2. EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES 273