STOCHASTIC

(k) Show that Theorem 1 is valid under the following relaxed assumptions on the value
function and investment allocation sets:
(i) Vn = f(X", r N ) is continuous in X N for all fixed r", where 0 < a < VN < 0 < oo.
(ii) There exists a sequence of strategies AN(e) e As such that
VN(AN(e))=f[AN{e),r N ] -+m*\r») = VN*
as e -+ 0, with associated s defined by VN(A„ (e)) = (1 — £) VN * + eVN.
Exercise Source Notes
Exercise 3 was adapted from Tobin (1965); Exercise 4 was adapted from Holt and Shelton
(1961); Exercises 5 and 6 were adapted from Keeney (1972); Exercises 8 and 9 were adapted
from Crane (1971); Exercise 10 was based in part on a private communication from G. Pye;
Exercise 14 was based on Bawa (1973); Exercise 16 was adapted from Stevens (1972);
Exercise 18 was based on Kelly (1956); Exercise 19 was adapted from notes written by
S. Larsson; and Exercise 22 was adapted from Brumelle and Larsson (1973).
COMPUTATIONAL AND REVIEW EXERCISES 675