Ergodic robust maximization of asymptotic growth

Abstract

We consider the problem of robustly maximizing the growth rate of investor wealth in the presence of model uncertainty. Possible models are all those under which the assets’ region E and instantaneous covariation c are known, and where the assets are stable with an exogenously given limiting density p, in that their occupancy time measures converge to a law governed by p. This latter assumption is motivated by the observed stability of ranked relative market capitalizations for equity markets. We seek to identify the robust optimal growth rate, as well as a trading strategy which achieves this rate in all models. Under minimal assumptions upon $(\mathit{E},\mathit{c},\mathit{p})$, which in particular allow for an arbitrary number of assets, we identify the robust growth rate with the Donsker–Varadhan rate function from occupancy time large deviations theory. We also explicitly obtain the optimal trading strategy. We apply our results to the case of drift uncertainty for ranked relative market capitalizations. Here, assuming regularity under symmetrization for the covariance and limiting density of the ranked capitalizations, we explicitly identify the robust optimal trading strategy.