STOCHASTIC

JAMES A. OHLSON
Assumptions A1-A3 are extremely stringent; Al requires that all moments
are finite, and, as will be shown in subsequent analysis, this is quite unnecessary.
Perhaps more important, even if all moments are finite and satisfy Al, it does
not follow that A2 and A3 are always satisfied for a utility function which is
everywhere differentiable an infinite number of times. In fact, not even A3
is implied by A2. As a first illustration, suppose U{W) = \ogW, and
Pr{K> W> 2} > 0 for every t, no matter how small. Although Al may well
be satisfied, it is still true that
j = o
is not defined for any t > 0. Clearly, A2 is a nontrivial restriction. An example
where A2 is true but A3 is not can also be easily provided. Consider the case
in which U(W) = — exp{ — cW), c> 0, and where F(W) is a distribution with
finite moments. Obviously, if F(0) = 0, then E \ U(W)\ < oo and A2 is satisfied.
However, there are distributions which yield quadratic utility as t -* 0,
but lim„^ao|£/ ( " ) (l)£(»'-l)7/i!| = oo so the infinite series in (5) will not
converge. 4 From both of these examples it is apparent that assumptions
A1-A4 are not particularly useful even for the cases in which U(W) or W or
both are bounded.
In motivating the development of a set of weaker assumptions than those
given so far (A1-A4), it is instructive to analyze the remainder term in expression
(3), EQ3. Now, for any n jg 3,
EQ3 = E Xc/ (0 (i) ((**--i)7n) + e„+i
i=3
Assuming the first two moments are of order 0(t), one weak necessary
condition for asymptotic quadratic utility is that E(W— 1)" = o(t) for some
n ;> 3; if E(W-1) 4 = 0(t), say, it is then hardly plausible that EQ3 = o{t).
More generally, it is desirable to state assumptions about F, such that these
assumptions imply that no "information" is contained beyond the first two
moments as / -> 0. The characteristic function of a random variable determines
the distribution uniquely; it is therefore appealing to develop conditions
on the moments such that the characteristic function is asymptotically
determined by the first two moments alone. Preferably, the latter should hold
even though higher order moments may not be finite. The following result
provides some intuitively obvious and useful conditions.
Theorem I Suppose £(^-1) and E{W-\) 2 are of order 0(0, and
* See Section II in this paper; there it is assumed the returns are log normally distributed.
224 PART III STATIC PORTFOLIO SELECTION MODELS