STOCHASTIC

MINIMAX POLICIES FOR SELLING AN ASSET AND DOLLAR AVERAGING 387
achieved by selling all of the asset which remains (i.e., x = 0). It is also shown that
an upper bound on £i(n, z) is a(T — t) — n for « £ (T — t)a. In particular it is
shown that L,(n, 0) = a{T — t) — n. That this is an upper bound follows from the
fact that L% is shown also to be a decreasing function of z.
THEOREM 2. (a) Li(n, z) is positive forn 0. Using these facts, 0 < X < 1, and
the just stated convexity inequality gives:
i,(n', z) £ X _1 £((n", z) > L,(n", z).
Since £«(n', z) > L,(n", z) for any n and n" such that 0 £ n < n" < a(T — t)
and for n < 0 from (12) it follows that L, is a strictly decreasing function of n for all
n < a {T — /) as was to be proved.
(c) It follows immediately from (14) that Lt is a decreasing function of z, given n,
because increasing the value of z enlarges the set of values on which the minimum with
respect to x can occur. L, need not be strictly decreasing in z. To see that
L,(n, 0) = (T — t)a - n for n < (T - t)a,
first note from part (b) above and (14) that:
i,(n, 0) = Max [L,+1(n - a, 0), £,+i(n + 6, 0)] = Lt+1(n - a, 0).
Using this recursion relation and (13) gives:
L,(n, 0) = LT(n - {T - t)a, 0) = (T - t)a - n.
This completes the proof of the theorem.
It remains to characterize the minimax variable regret with respect to time. In the
next theorem it is shown that Lt{n, z) is a decreasing function of time, given n and z.
It is a strictly decreasing function of time as long as the value of n is not so large as
to preclude the previous maximum value of the price being exceeded before time is up.
The proof requires the following trivial lemma.
LEMMA 1. Letf(x) = Max„SiSI\fi(x)]andg(x) = MaxoSiii\gi(x)].Iffi{x) > gt(x)
foroUi and x, fhenf(x) > g(x) for all x.
3. MODELS OF OPTION STRATEGY 585