Rapid paths in von Neumann–Gale dynamical systems

Abstract

The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such 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 von Neumann–Gale dynamics is played by a special class of paths called rapid (they maximise properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades.

Keywords:

• random dynamical systems
• convex multivalued operators
• von Neumann–Gale model
• rapid paths
• Dynkin's and Radner's dualities

AMS Subject Classification:

• 37H99
• 37H15
• 91B62
• 91B28

Acknowledgements

Financial support from the grant NSF DMS-0505435, the State of Missouri Research Board, the University of Missouri-Columbia Research Council, and the Manchester School Visiting Fellowship Fund is gratefully acknowledged.

Notes

¹ A closed set G(ω) in a metric space is said to depend measurably on ω if the distance to this set from each point in the space is a measurable function of ω.