STOCHASTIC

390 GORDON PTE
Inspection of this last expression shows that zT-i(n) is given by
2?_I(TO) = (a - n)/(a + b) 0 S n £ a
— 0 n > o.
Thus, 2r_i(n) is a decreasing function of n as required. Using this it is also easily verified
that LT(a — n, z) — az = LT(n + b, z) + bz for all n and all 2:0 £ z £ zr_i(«)
as required. This completes the proof.
It might be conjectured that the critical values are a decreasing function of t as
well as n. This, however, is not true. A counterexample is provided by the minimax
policy calculated for the example in the following section. On the basis of examples it
appears that the critical values are much less sensitive to t than n except near T.
This, taken together with Theorem 5, has the following implication.
After some of the asset is sold initially, little more will be sold if the price rises or
stays the same. In this case n will remain equal to zero. On the other hand, more of
the asset will be sold as soon as the price drops because then n will increase. In general,
for a sequential pohcy, more is sold on price dips than rises because the former increase
n while the latter do not. An example for a simple case is provided in the next
section.
4. An Example of a Sequential Policy
As a simple example of an arithmetic random walk consider the following process.
Suppose that each period the price can either go up by one unit, down by one unit or
stay the same. This means that a = 6 = 1 and that n can take on only integer values.
Suppose furthermore that there are three points of time available to sell the asset
before it must all be sold so that T = 3. When t = 2 and n S 1 it is already known
from Theorem 2(a) that the minimax policy is to sell all of the asset which is left. For
t = 2 and n = 0, (13) and (14) give the following:
L2(0, z) = MinoS*s, Max [L3( — 1, x) — x, Lj(l, x) + x]
= MinoS»s, Max [1 — x, x]
= 1 — z z S | x = z z •& \
= J z>\ = i « > i-
Thus, the critical value, Z2*(0), is f. Any asset which is held above this amount should
be sold. If less than this amount is held none should be sold.
Using this result, the minimax strategies for t = 1 and n = 0, 1 can next be derived.
LxtO, z) = Min0 I
ii(l, z) = Min0SlS2 Max [L2(0, x) — x, Li(2, x) + x]
= Minos*s, Max IL _ * * = \ ,x\
= l-2z « i l x* = z zfij
588 PART V. DYNAMIC MODELS