The trade-off between growth and risk in Kelly's gambling and beyond

We study a generalization of Kellys horse model to situations where gambling on horses other than the winning horse does not lead to a complete loss of the investment. In such a case, the odds matrix is non-diagonal, a case which is of special interest for biological applications. We derive a trade-off for this model between the mean growth rate and the volatility as a proxy for risk. We show that this trade-off is related to a game-theoretic formulation of this problem developed previously. Since the effect of fluctuations around the average growth rate is asymmetric, we also study how the risk-growth trade-off is modified when risk is evaluated more accurately by the probability of the gambles ruin.