STOCHASTIC

EXPANDING BUSINESSES OPTIMAL 649
It is easily shown that this is equivalent to choosing the x ^ so that they satisfy
X, = E
Ix
Hence, we may verbally state this investment policy as: let the investment proportion in alternative
i be equal to the expected proportion of the return coming from alternative i . It is
interesting, as well as somewhat gratifying, that this policy is one of diversification; that is,
in general, this policy is one of dividing S^j between a number of alternatives.
Finally, as far as the mathematics of the situation is concerned, our main tool is the
ever-reliable martingale theorem.
MATHEMATICAL FORMULATION AND PROOFS
Let X„ = (x., ..., X ) be the investment proportion vector for the N" 1 period, i. e.
n
X j is the proportion of Sj^ invested in alternative i; so using the notation V,. =£ X. r, we
have
S N+1 = S N V N '
Now let Rjj-l be the outcomes during the first N-l investment periods,
Rjj-1 = (fj(.i> • • •, fj) . We consider two competing sequences of investment policies
CX|, ^2> • • •) anc ' (*l> ^2' '' •) sucn tna ' un( ^ er eacn we start with the same initial fortu
Now Xj^, Xj^ may depend in an arbitrary manner on RJJ.J as may the distribution of r^. Let
SN be the fortune under (X^ ...) and S& the fortune under (T.J,
defined as that investment proportion vector which maximizes
E(1 °e ^IW
V N= 2>IV S 5 °.Z*i
We need some uniform integrability condition, and we will assume that there are constants
a, B such that
0 <