STOCHASTIC

or less than u; either do nothing or move the cash balance to U whenever it
is between u and u + ; either do nothing or move the cash balance to D whenever
it is between d~ and d; and move the cash balance to some (unknown) value
between u + and d~ whenever it reaches this region. The simple («, U; D,d)
policy is a special case of this more complex policy; whether or not it is optimal
in the general case appears to be an unsolved problem. Part (j) of Exercise
ME-11 presents a special case for which optimality of the (w, U; D,d) policy
is known to be true. Girgis (1969) proved the optimality of such a simple
policy when at least one of the fixed transfer costs is zero. Recently, Vial (1972)
established the optimality of the simple policy for the general case in a
continuous-time model, with changes in the cash balance level being given by
a diffusion process.
For additional information on the cash balance problem, the highly readable
and informative book by Orr (1970) is recommended. It discusses all of the
mathematical cash balance models up to 1970, and, in addition, presents
empirical analysis, discussions of institutional constraints, and economic
consequences of various cash balance models. Except for Miller and Orr
(1968) and one chapter by Orr (1970), most existing cash balance models
have treated the problem in a "two asset" setting, where in addition to idle
cash, there exists a single alternative earning account. A more realistic model
may require the existence of several alternative earning accounts, each having
its own "risks" and transfer costs. The ultimate cash balance model would
thus combine portfolio choice with inventory theoretic considerations. A
tentative step in this direction has been taken by Eppen and Fama (1971),
who treat the cash balance problem in a "three asset" setting, using stochastic
dynamic programming. For an interesting treatment of sequential policies for
bank money management in a minimax regret framework, see Pye (1973).
For further results and information on the cash balance problem, the reader
may consult Daellenback and Archer (1969), Frost (1970), Heyman (1973),
Homonoff and Mullins (1972), Orgler (1969, 1970), Porteus (1972), and
Porteus and Neave (1972).
V. The Capital Growth Criterion and Continuous-Time Models
The papers by Breiman and Thorp are concerned with Kelly's capital
growth criterion for long-term portfolio growth. The criterion states that in
each period one should allocate funds to investments so that the expected
logarithm of wealth is maximized. Hence the investor behaves in a myopic
fashion using the stationary logarithmic utility function and the current
distribution of wealth.
Kelly (1956) supposed that the maximization of the exponential rate of
growth of wealth was a very desirable investment criterion. Mathematically the
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