Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators
A multivalued stochastic dynamical system is defined here by means of his paths or trajectories -- instead of using the system of stochastic differential equations, for example -- which are supposed to be (generated by) the sequences of random vectors \(\{x_t\}\) such that \(x_t \in A_t(x_{t-1})\); the mappings \(x\mapsto A_t(x)\), \(t=1, 2,\dots\), are set-valued nonlinear operators assigning a nonempty set \(A_t(x)\subseteq X_t\) to each \(x\in X_{t-1}\), where \(X_t\) is the cone of nonnegative elements in the set of essentially bounded \(F_t\)-measurable functions defined on some probability space \((\Omega,F,P)\) with filtration \((F_t)\) and with values in the \(n\)-dimensional Euclidean space. In the deterministic case, dynamical systems of the above-mentioned type reduce to products of homogeneous convex multivalued mappings defined on the cone of nonnegative \(n\)-dimensional vectors. A profound theory of such mappings has been developed by many authors. The present work may be regarded as a step towards the construction of a stochastic variant of this theory. The main goal of this paper is to single out and investigate a class of paths which grow faster in a certain sense than other paths. They are called rapid. Infinite rapid paths are of primary interest, since they bear important information about the asymptotic properties of the dynamical system in question. However, the present article is basically devoted to the study of finite rapid paths. Infinite ones will be examined in detail in the author's next publications. Here only some of their general properties are discussed. The main purpose of the paper is to lay the ground work for further study. Central results of the article are existence and characterization theorems for finite rapid paths. These theorems yield methods for constructing such paths.
scientific article; zbMATH DE number 1230458