STOCHASTIC

42 LELAND
than or equal to \Ki. The second RHS term can be written, for any
8 > 0, as
Pr(0 < j8 < S) £[t/(A)8) - U(0) | 0 < /3 < S]
+ /V(£ > S) £[t/(Aj3) - 1/(0) | /3 > 8]. (2)
Consider the first term of (2). From U.2, we have
Pr(0 sg & < 8) £[C/(A0) - 1/(0) | 0 < 0 < 8]
< AiV(0 < j8 < 8) £[j8 | 0 < jS < 8] l/'(0)
= Atf(S), say, where /C(S) > 0 and /C(S) ->- 0 as 8 ->- 0.
Choose 8, so that K(8) < —(J) Kx . To consider the second term of (2),
we need the following
LEMMA. If U(c) satisfies U.\, U.2, and [/.3a: Iim U'(c) = 0, then
lim[t/(c) - U(0)]/c = 0.
C-»cc
Proof. Note U.2 implies [U(c) — U(0)]/c is monotonically decreasing
in c; C/.l implies it is always nonnegative. Therefore, it must approach
a limit. Assume it approaches a limit -q > 0. Then [U(c) — C/(0)] ^ ijc,
or, since U'(c) approaches a limit by C/.l and £/.2, lim U'(c) ^ r;, a
contradiction. Therefore, lim[[/(c) — U(0)]/c = 0.
e-»oo
The lemma implies that, given any y > 0, there exists a A(y) such that,
for c > A(y) 8, then U(c) - C/(0) < yc. Therefore,
Pr(P > 8) E[U(Xp) - 1/(0) | j8 > 8] ^ Ay£[j8 | /3 > 8] i>r(/3 > 8) (3)
for all A > A(y). Note that the assumption E[fS] < oo implies
£[j8 113 > 8] < oo.
From (3), we may choose y so that Ay£[£ | j3 > 8] PrflS > 8) < -(!)#, A.
for all A > A(y). Then, for all A > A(y),
V{Xy) - V(0) < [AT, - (J) Kt - (J) if,] = (J) tf,A < 0,
as required. The rest of the proof continues as in Theorem I.
PROPOSITION IV. A*( y) is a continuous function ofy, yeY.
Proof. We have shown for each yeY, there exists a A* such that
V{X*y) = k= V(0). Take any yeY, and notice VK., Vy exist and are
continous in a neighborhood of A*, y. Vk» is negative, because V is strictly
PART III. STATIC PORTFOLIO SELECTION MODELS