STOCHASTIC

Note that (f) and (g) imply the existence of the limit/(;') = lim„_„o f„(i). Now let
G(;) = £(/) + « t fU+k)Pk,
and define
D = min{j| AG(/) > -c„}, U= max{j| AG(J) > ct).
(h) Show that the return points U„,D„ converge to U,D and that the latter are finite.
[Hint: Show that
Show that
where
Now let
K AG„0) § AL(J) + a AG„_, (j- K).
0 < j < D„ => GnU) S £ «"ch - « M+l cu
M = max{/n | m g //#}.
n = min{w| Y. a*ch —a m + 1 c„ > c,,}.
fc = 0
Note that « < oo, and show that D„ ^ (« + 2)A^for« > max(n,,n2).]
(i) Show that/(0 satisfies the functional equation
/(») = min
j
/, J + L(;) + a £ f D, decrease
the cash balance to D; (2) if i < U, increase the cash balance to U; (3) if U g i' ^ D, do
nothing.
Assume now that there are also fixed transfer costs for changing the cash balance. Thus
(Ku + Cu(j-i) if j>
t, J = 0 if j =
\Kd + cd(i-j) if j <
where K„ and Kd are greater than zero. By analogy with inventory theory, it might be suspected
that the optimal policy is of the form («, U;D,d): (1) if i > d, decrease the cash balance
to D; (2) if i < u, increase the cash balance to U; (3) if u g i g d, do nothing. (Whether or
not this is true in general appears to be unknown.)
Assume now that the only transfer costs are the fixed costs; that is, assume that c„ =
c„ = 0.
(j) Show that the optimal policy is of the form («, U; D, d) with U' = D, that is, show
that the two return points are identical.
(k) If in addition cb = cv and K„ = Kd, and if pk is symmetric (pk=P-k, \k\ ^ K),
show that d= —u and D = (7=0. {Note: This problem says that for a symmetric cash
balance problem involving only fixed transfer costs, the optimal policy is symmetric]
(1) What is the form of the optimal policy for cp = oo ?
12. Refer to the article by Pye concerned with call options on bonds.
(a) Suppose that the p's have a general distribution F(p l ,p 2 , ...,p T ), where p' refers to
the value of p in period i. Formulate a T-period stochastic program that describes the
decision problem, and develop an appropriate dynamic programming functional
equation.
Develop solution methods for the cases when:
(b) the p' are independent and identically distributed,
MIND-EXPANDING EXERCISES 689