How do stochastic systems achieve maximum growth? This paper examines set-valued random dynamical systems defined by convex homogeneous stochastic operators. These operators transform elements of a cone contained in a space of random vectors into subsets of the cone. The study focuses on rapid paths of such dynamical systems, which maximize (appropriately defined) growth rates at every time period. The work addresses questions of existence, uniqueness, and asymptotic behavior of infinite rapid trajectories. By investigating these mathematical properties, the authors provide insights into the long-term behavior of such systems. Motivated by problems related to stochastic models of economic growth, this research offers a rigorous mathematical framework for analyzing systems characterized by uncertainty and set-valued dynamics. The results contribute to the theoretical foundation for understanding economic growth and other complex dynamic processes.
This article is aligned with the scope of Stochastics and Dynamics, which focuses on mathematical aspects of stochastic processes and dynamical systems. The paper's investigation of rapid growth paths in convex-valued random dynamical systems contributes to the journal’s scope by advancing the mathematical understanding of complex systems with applications in various fields, including economics.