Generalized Compounding and Growth Optimal Portfolios Reconciling Kelly and Samuelson

We generalize the Kelly criterion and the growth-optimal portfolio (GOP) beyond log-wealth maximization. We show that time-change models require compounding algebras and GOPs that do not coincide with maximization of the expected log of wealth. In the variance gamma (VG) and the normal inverse Gaussian (NIG) models the generalized GOP concepts mimic well-known utility models, namely power utility and the mean variance approach, with a parameter that, in both cases, is the variance of the stochastic clock. When the variance of the stochastic clock goes to zero, the model retrieves the standard Kelly criterion and GOP is the expected logarithm of wealth maximization.