Dynamics under uncertainty

TL;DR: In this article, a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents is developed, based on recent results of Bismut [2, 3] concerning the characterization of the extrema of stochastically variational problems over a finite horizon and on their own investigations of the stability properties of the equations of dynamic economics.

Abstract: THIS PAPER IS a preliminary investigation of dynamics under uncertainty. We attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents. The analysis builds on recent results of Bismut [2, 3] concerning the characterization of the extrema of stochastic variational problems over a finite horizon and on our own investigations [6, 7, 20, 21] of the stability properties of the equations of dynamic economics.2 We consider a class of discounted infinite horizon maximum problems. While it is convenient to pose the basic economic problem as a stochastic control problem, to obtain the full benefit of Bismut's elegant characterization of a maximizing process it is convenient to transform this problem into an equivalent stochastic variational problem along the lines indicated by Rockafellar [27] in the deterministic case and generalized by Bismut [2] to the stochastic case. Within this framework we show that the idea of a competitive path introduced in the continuous time deterministic case in [21] generalizes in a natural way in the case of uncertainty to a competitive process. We show, under a concavity assumption on the basic integrand of the problem, that a competitive process which satisfies a transversality condition is optimal under a discounted catching up criterion (Section 2). In Section 3 we examine the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price process generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie [22] who first recognized the importance of this property in the deterministic case. We show that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The earlier convergence result extensively studied in the deterministic case thus continues to hold in the stochastic case. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem. Section 4 examines the long-run behavior of the probability measure associated with a competitive process. We give conditions under which a McKenzie competitive process is a Markov process with an invariant probability measure and