An algorithm for maximizing expected log investment return

Let the random (stock market) vector X 2 0 be drawn according to a known distribution function F(x), x E R ". A log-optimal portfolio b* is any portfolio b achieving maximal expected log return W* = sup,, E In b'X, where the supremum is over the simplex b 2 0, Cr, b, = 1. An algorithm is presented for finding b*. The algorithm consists of replacing the portfolio b by the expected portfolio b', b; = E(b, X,/b'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'll b) between the current and updated portfolios. Thus the algorithm monotonically improves the return W. An upper bound on W* is given in terms of the current portfolio and the gradient, and the convergence of the algorithm is established.