can be an optimal solution for the non-linear programming problem only if there exist

(8) |

where

But what if an investor were to use causal investment strategy? We can prove that with probability 1, the conditionally log-optimal investor will not do any worse than any other investor who uses a causal investment strategy. Let

(9) |

be the wealth after

(10) |

be the maximal doubling rate and let

(11) |

that is,