STOCHASTIC

382 GORDON PTE
strategies. Minimax policies will tend to have a reinforcing effect on prive movements.
In §6 the sequential and nonsequential minimax policies are derived for the case where
repurchases of the asset are allowed so that the problem becomes like portfolio seleccion.
2. Nonsequential Policies
Assume that there are T points of time at which some of the asset can be sold. The
first such point is numbered zero and the last T — 1. After each of these points and
before the next, the price of the asset changes according to the assumed stochastic
process. When the price has changed after T — 1, any of the asset remaining is automatically
sold at time T. In the nonsequential case a decision is made at time zero as to
how much to sell not only at time zero but at each future point of time as well. In
the sequential case a decision need not be made as to how much to sell at a particular
point until the preceding realizations of the stochastic process are known.
Let the amount of the asset held before sales are made at t be z(. After sales are
made at t the amount held is z,+i. Therefore, the amount sold at t is z, — z,+l or
— Azi+i. Since none of the asset can be repurchased the z, must be nonincreasing with
t (i.e., Zi+i £ zi). For convenience the amount of the asset to be sold initially will be
set equal to one (i.e., zo = 1). Since all of the asset must be sold by T, zT+i = 0. The
asset is taken to be perfectly divisible.
The regret for a given policy and time series of prices is equal to the difference of
two quantities. The first is the largest amount which could be obtained using the
optimal feasible policy. Since purchases are prohibited, this is simply to sell all of the
asset at the highest price. Let pt be the price at which the asset can be sold at time t.
Let t* be the time at which the maximum price occurs. Below it will be convenient to
make use of the following identity for p,.
(1) Pf = Po + £{* Ap,.
When t* = 0 in (1) the summation on the right-hand side is taken to be equal to zero.
To obtain regret, the amount obtained using the given policy must be subtracted from
this maximum price. The amount obtained is given by the following expression:
(2) iy +1 - P