Abstract
This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered here are supposed to be unbounded, and the obtained strategies are of non-stationary closed-loop type. The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing three algorithms for the purpose of replicating the results. There, the reader can find a result on the interchangeability of limits in a Dirichlet problem.
Highlights
The aim of pollution accumulation models is to study the management of some goods to be consumed by a society
The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing how to replicate the theoretical results of the work
Constrained optimal control under the discounted and ergodic criteria was studied in the seminal paper of Borkar and Ghosh, the work of Mendoza-Pérez, Jasso-Fuentes, Prieto-Rumeau and Hernández-Lerma, and the paper by Jasso-Fuentes, Escobedo-Trujillo and Mendoza-Pérez [14]. These works serve as an inspiration to pursue an extension of their research to the realm of non-stationary strategies. This is not the first time that the problem of pollution accumulation has been studied from the point of view of dynamic optimization, this paper contributes to the state-of-the-art by adding constraints to the reward function, and by taking into consideration a finite-time horizon
Summary
The aim of pollution accumulation models is to study the management of some goods to be consumed by a society. These works serve as an inspiration to pursue an extension of their research to the realm of non-stationary strategies This is not the first time that the problem of pollution accumulation has been studied from the point of view of dynamic optimization (for example, [15] uses an LQ model to describe this phenomenon, [16] deals with the average payoff in a deterministic framework, [17,18] extend the approach of the former to a stochastic context, and [19] uses a stochastic differential game against nature to characterize the situation), this paper contributes to the state-of-the-art by adding constraints to the reward function, and by taking into consideration a finite-time horizon. It is not difficult to see that if F meets Hypothesis (H2a)–(H2c), so do the social welfare, the cost rate and the running constraint functions
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