Log-optimal and rapid paths in von Neumann-Gale dynamical systems

Abstract

Von Neumann-Gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of such systems is played by a special class of paths (trajectories) called rapid: they grow over each time period [t−1,t] in a sense faster than others. The paper establishes existence and characterization theorems for such paths showing, in particular, that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid. The proof of this result is based on the methods of convex analysis in spaces of measurable functions. The study is motivated by the applications of the theory of von Neumann-Gale dynamical systems to the modeling of capital growth in financial markets with friction.