STOCHASTIC

116 HOWARD M. TAYLOR
the investor would purchase under these conditions with j'S 1 periods to go if
y < E[P,-i(Y,-_!) | Yj = y); i.e. if the price y of buying immediately were less
than the expected minimum price if an immediate purchase is not made. One
has Po(y) = min \y, po) for all y, and, knowing the optimal procedure in any
period, we have Pj(y) = min \y, E[Pj-i( Yj-i) | Yj = y}}, for all y and j & 1.
Let Hj(y) = Pj(y) - V so that Hj(y) = min {0, EIH^Y^,) | Yj = y] +
E[Yj-i — Yj | K, = i/]}. In terms of the new function, if Yj - y and Hj{y) = 0,
we would immediately purchase on the market. More exactly, if Jj(y) =
E[Hi-i(Yj-i) | Yt = y] + £[(K3_i - Yj) \ Yj = j/], then we would purchase
immediately if Jj(y) > 0; we are indifferent if Jj(y) = 0; and we wait at least
one more day if Jj(y) < 0.
By assumption (2.3) E\Y,-\ — Yj | Yj = y] is a noDincreasing function of y.
Also, flo(2/) = min {0, po — y] is a nonincreasing function of y, and by assumption
(2.2) and subsequent remarks, if ff,_i(?/) is nonincreasing in y, then so
also is E[Hj-i(Yj-i) ] Yj = y\. Since the sum of nonincreasing functions and the
minimum of 0 and a nonincreasing function are also nonincreasing, we have
Jj(y) and H;(y) are nonincreasing in y for ally £ 0. There is an additional
monotonicity. For y S p0, H0(y) — 0 and Hi(y) £ 0 so that H,(y) g He(y)
in this case. For y > po,
H,(y) g E[H„(Yo) \Y1 = y) + E[Y„ - Y1\Y1 = y]
= / (p„ - t) dF(t | y) + f (t-y) dF(t \ y)
pQ
J-tn
dF(t \y) + (t-y) dF(t \ y) £ (p„ - y) = #„(*/).
Thus HI(J/) g Ho(y) for all i/. Now assume Hj(y) S Hj-,(y) for all i/. Then
£[#,(7,) | Yi+1 = y] = j Hj(t) dF(t \y)i J Hj-r(t) dF(t \ y)
= E[Hj-i{Yj-i)\Yi
= y\.
Similarly,
E\Yj - Ym | Yi+l = 2/] = / (t-y) dF(t | y) = E[Y^ - Yj | Yj = y].
By addition Jj+i(y) § •/>(#), and then Hi+i(y) = min {0, ^y+i(y)| S min
JO, J,(y)} = Hj(y). Thus by induction, Hj(y) and Jj(y) are nonincreasing in j
for all y.
We are now ready to draw our first two conclusions. Let ao = po, and for,;' S 1
let Oj = sup |^: Jj(y) > 0) if the set is non-empty, and a3- = — » otherwise.
With j periods to go and a current price of Yj, an optimal strategy is to purchase
immediately if Yj < a,- and to wait otherwise. From the monotonic properties
of Jj(y) m j a °d V we have av( = Po) £ at ^