STOCHASTIC

410 MERTON
of return JX. The interpretation of this result is that as ft increases for a
given a, the probability increases that future "ct's" will be more favorable
relative to the current a, and so there is a tendency to hold more of one's
current wealth in the risk-free asset as a "reserve" for investment under
more favorable conditions.
The last type of price mechanism examined differs from the previous
two in that it is assumed that prices satisfy the geometric Brownian
motion hypothesis. However, it is also assumed that the investor does
not know the true value of the parameter a, but must estimate it from
past data. Suppose P is generated by equation (119) with a and a constants,
and the investor has price data back to time •—T. Then, the best estimator
for a, &{t), is
1 r* dP
(130>
^ = TT-rL-P'
where we assume, arbitrarily, that