An algorithm for maximizing expected log investment return

Let the random (stock market) vector
X \geq 0
be drawn according to a known distribution function
F(x), x \in R^{m}
. A log-optimal portfolio
b^{\ast}
is any portfolio
b
achieving maximal expected
\log
return
W^{\ast}=\sup_{b} E \ln b^{t}X
, where the supremum is over the simplex
b \geq 0, \sum_{i=1}^{m} b_{i} = 1
. An algorithm is presented for finding
b^{\ast}
. The algorithm consists of replacing the portfolio
b
by the expected portfolio
b^{'}, b_{i}^{'} = E(b_{i}X_{i}/b^{t}X)
, corresponding to the expected proportion of holdings in each stock after one market period. The improvement in
W(b)
after each iteration is lower-bounded by the Kullback-Leibler information number
D(b^{'}\|b)
between the current and updated portfolios. Thus the algorithm monotonically improves the return
W
. An upper bound on
W^{\ast}
is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.