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Side Information

What happens if we do not have the correct information to select our portfolio? This is equivalent to choosing bg*, corresponding to probability density g(x), while bf* (corresponding to probability density f(x)) is the correct one. We can prove, given that $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ i.i.d.  f(x), that

\begin{displaymath}\Delta W = W(\mathbf{b}_f^*, F) - W(\mathbf{b}_g^*, F) \leq D(f \vert\vert g).\end{displaymath}

Which tells us that the increase $\Delta W$ in doubling rate due to side information Y is bounded by the mutual information between the side information Y and the stock market X.

\begin{displaymath}\Delta W \leq I(\mathbf{X};Y).\end{displaymath}



Magnus Bjornsson
1998-05-12