STOCHASTIC

648 L. BREIMAN
We have discussed the undesirability of the above criterion from the point of view of
ultimate ruin; that is, from the long-run point of view. In our opinion, most non-speculative
investment policies implicitly take this view. In order to formalize this criterion, we assume
that we start with an initial fortune SQ, and that SN ., our fortune at the beginning of the
N+l s ' period, consists of the total returns from our Investments of the N" 1 period; i. e. everything
that we make during the N* n period is plowed back into investments during the N+l s *
period. This assumption is made in order to be able to compare the asymptotic behavior of
SN under different investment policies, and is approximately valid during the initial expansion
period of business firms and in investment holdings where the dividends taken out of the system
are small compared with the total dividends. Any well-formulated rule for the removal of
funds from the system will evidently enable an asymptotic analysis of SxjJ we use the above
rule, however, for simplicity and illustration.
Concerning the Investment alternatives, we assume very little; they may vary arbitrarily
from period to period and the distribution of the pay-off vector rjj for the
Nth
period
may be conditioned in any way by the past. Perhaps our most drastic assumption is that any
division of funds between the various alternatives is an allowable investment policy. This is
unrealistic in the many situations where there is either an upper limit to investment, or a
lower limit, or both.
Naturally, our investment policy for the N" 1 period may depend both on S„ and on the
outcomes of the past. Now let S^ be the fortune of the beginning of the N" 1 period under one
sequence of investment policies and SN the fortune of the same time under a competing sequence
of investment policies. We will say that the former sequence is inadmissible if there is a
fixed number a > o such that for every e > o there exists a competing sequence such that on
a set of probability greater than a - e
lira „ sup - r
and, except for a set of probability at most e,
Roughly, the sequence leading to S», is inadmissible if for every e > o there is a competing
sequence that is asymptotically infinitely better with probability at least a-e and with, at most,
probability e of being worse.
Restricting ourselves to admissible sequences of policies, we domonstrate that there
is a unique admissible sequence which is distinguished by the fact that every admissible
sequence must be asymptotically close to it. This sequence is found as follows: let
ll ..., \n be the proportions of Sfj that we invest in alternatives 1, ..., N, then choose
the \s so as to maximize
! N
SN
SN
Elog (£>i rjj , Ipo, £»j
594 PART V. DYNAMIC MODELS