ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-06262014-124018


Tipo di tesi
Tesi di laurea specialistica
Autore
BARTALONI, FRANCESCO
URN
etd-06262014-124018
Titolo
Intertemporal utility maximization problems with state constraints: existence theorems and dynamic programming
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
correlatore Prof. Gozzi, Fausto
relatore Prof. Acquistapace, Paolo
Parole chiave
  • optimization
  • state constraints
  • dynamic programming
  • utility
Data inizio appello
18/07/2014
Consultabilità
Completa
Riassunto
The main object of the present work is the study of a control problem, introduced and studied by D. Fiaschi and F. Gozzi, arising in the context of economic theory of growth. As the object of such studies is the analysis of the optimal trajectories defined by the state constraints, good existence results as well as handy sufficient and necessary conditions are specially needed.
Due to the generality of the hypotheses on the data, this problem is not straightforward. Indeed, there is no evidence of similar results in the existing literature, where certain growth conditions on the dynamics and some boundedness assumptions on the control set are usually required. We assume instead that the controls are only locally integrable, and the production function is increasing, unbounded and not globally concave. The use of non-concave production functions has a crucial role in the recent economic growth theory. However the technical difficulties arising in applying the classical analytical methods under these non-classical assumptions have led many authors to drastic simplifications in their analysis; therefore the results often lack a sufficient degree of generality.
The framework of our model is that of intertemporal infinite-horizon utility maximization with dynamic programming methods; for generality purposes, the case of finite-horizon cost minimization is also studied, with special emphasis on the notion of viscosity solution to a Hamilton-Jacobi-Bellman equation. This is done in the first part of the thesis. In both cases, we show that the value function is a viscosity solution of the proper HJB equation, and we give also a uniqueness result in the finite horizon case.
For all these reasons, this thesis is conceived as a first step of a more ambitious program that we hope to carry on in the near future.
File