STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
The/j are known in closed form in only very special cases; hence the most
convenient way to study the stable family utilizes the characteristic function
^,(l)s Ee" y = f" e""f(y) dy where i = yf^l.
J— CO
The log characteristic function for a symmetric stable distribution
F,tf,;&,Sl,0,a)is
ln^l(/J) = i|Jr,-SJ|rl|«.
Asset 0 is riskless and returns |0 with certainty and exhibits constant
returns to scale so that if x0 is invested, the return is |0 x0. x0 < 0 corresponds
to borrowing at the risk-free rate. It is assumed that the investor may borrow
or lend any amount at the constant rate of |0. This asset may be considered
to have the degenerate stable distribution F0(£0; |0,0,0, a).
It is supposed that the investor wishes to choose the xt to maximize the
expectation ofa utility functions of wealth w= t,'x, where