A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of L (±α) -divergences. It includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the L-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. They admit natural generalizations of exponential family that are closely related to the α-family and q-exponential family. In particular, the L (±α) -divergences of the corresponding potential functions are Rényi divergences. Using this unified framework we prove that the induced geometries are dually projectively flat with constant sectional curvatures, and a generalized Pythagorean theorem holds true. Conversely, we show that if a statistical manifold is dually projectively flat with constant curvature ±α with α > 0, then it is locally induced by an L (∓α) -divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.

No abstract

Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The “universality” refers to the fact that this result is model‐free , that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.