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## Scuola Dottorale in Scienze Astronomiche, Chimiche, Fisiche,

### Citations

3924 |
Dynamic Programming.
- Bellman
- 1957
(Show Context)
Citation Context ...ynamic Programming Principle A very powerful approach to this optimal control problem is to use the dynamic programming, which was first introduced in 1954 by Bellman with his principle of optimality =-=[5]-=- (see also [6]) which says that 1.2 Optimal control theory 8 an optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an o... |

1110 |
An Intertemporal Capital Asset Pricing Model
- Merton
- 1973
(Show Context)
Citation Context ...problem into a sequence of simpler minimization (or maximization) problems. The Bellman equation was first applied in engineering, then it became an important tool in economic theory (see for example =-=[44]-=- for a first and celebrated economic application of a Bellman equation, other examples can be found for instance in [7, 41]). To apply the dynamic programming approach to this model problem we introdu... |

872 |
Stochastic differential equations. An introduction with applications
- Øksendal
- 2000
(Show Context)
Citation Context ... +∞ 0 `(yεx(t), α)e −λtdt ] Under suitable conditions of the data, vε happens to be a C2 solution of (1.13) and this follows from the Dynamic Programming Principle and It o's stochastic calculus (see =-=[27, 45]-=- for more details). Moreover it can be proved, as was natural to expect, that when the randomness parameter ε goes to 0 the value function vε converges to the value function v, viscosity solution of (... |

645 |
The economics of exhaustible resources
- Hotelling
- 1931
(Show Context)
Citation Context ... The oil production framework A fairly typical example of mean field game is that of the production of an exhaustible resource by a continuum of producers. We know from a seminal article by Hotelling =-=[32]-=-, published in 1931, that there is a rent involved in the production of an exhaustible resource, but it is interesting to examine this in greater depth in a competitive situation and to understand the... |

550 |
Viscosity solutions to Hamilton–Jacobi equations
- Crandall, Lions
- 1983
(Show Context)
Citation Context ...ion (1.7) F (x, u(x), Du(x)) = 0 x ∈ R where Ω is an open domain of RN and the Hamiltonian F : R×R×RN → R is continuous. The term viscosity solutions first appear in a work of Crandall and Lions (see =-=[18, 19]-=- and, for further details, [17, 33]) in 1981, but the definition of solution had actually been given earlier by Evans [21] in 1980. Definition 1.6. A function u ∈ C is a viscosity subsolution of (1.7)... |

479 | Nonlinear programming
- Kuhn, Tucker
- 1951
(Show Context)
Citation Context ...ions that will be applied later in the elaborate. Moreover we recall a well known theorem, which is also used in the following. 1.1.1 Karush-Kuhn-Tucker conditions The Karush-Kuhn-Tucker conditions 1 =-=[34, 37]-=- are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach t... |

247 |
Some properties of viscosity solutions of Hamilton–Jacobi equations
- Crandall, Evans, et al.
- 1984
(Show Context)
Citation Context ...∈ R where Ω is an open domain of RN and the Hamiltonian F : R×R×RN → R is continuous. The term viscosity solutions first appear in a work of Crandall and Lions (see [18, 19] and, for further details, =-=[17, 33]-=-) in 1981, but the definition of solution had actually been given earlier by Evans [21] in 1980. Definition 1.6. A function u ∈ C is a viscosity subsolution of (1.7) if, for any ϕ ∈ C1, one has (1.8) ... |

227 |
P.: Dynamic Programming and Stochastic Control
- Bertsekas
- 1976
(Show Context)
Citation Context ...gineering, then it became an important tool in economic theory (see for example [44] for a first and celebrated economic application of a Bellman equation, other examples can be found for instance in =-=[7, 41]-=-). To apply the dynamic programming approach to this model problem we introduce one of the most important concept of the theory. We define the value function (1.4) v(x) = inf α∈A J(x, α) where A denot... |

223 |
Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations
- Bardi, I
(Show Context)
Citation Context ...ables that minimize the cost functional. In this chapter we will present a model problem to focus on the main concept of the optimal control theory. Most of the contents can be founded for example in =-=[47, 4, 23]-=-. 1.2.1 A model problem: the infinite horizon discounted regulator Consider for example a control system governed by the state equation (1.2) { y′(t) = f(y(t), α(t)) t > 0 y(0) = x Here the control α ... |

156 |
Minima of Functions of Several Variables with Inequalities as Side Constraints
- Karush
- 1939
(Show Context)
Citation Context ...ions that will be applied later in the elaborate. Moreover we recall a well known theorem, which is also used in the following. 1.1.1 Karush-Kuhn-Tucker conditions The Karush-Kuhn-Tucker conditions 1 =-=[34, 37]-=- are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach t... |

151 |
Computational Methods in Optimization.
- Polak
- 1971
(Show Context)
Citation Context ...s implementation 55 Conversely, using an interior penalty function, we can take a function ρ̄ ∈ BUC(Q) which is nonnegative and increasing as the constraint boundary ∂Q is approached (see for example =-=[46]-=- for further details). Then we define the running cost ¯̀(x, a) := `(x, a) + ερ̄(x) and cost functional J̄ and value function ū in the usual way. It is easy to show that the value function satisfies ... |

139 | Political Foundations of the Resource Curse.
- Robinson, Torvik, et al.
- 2006
(Show Context)
Citation Context ...nagement of resources, or weak, ineffectual, unstable or corrupt institutions. We start our work from a model, which is halfway between economy and politics, developed by Robinson, Torvik and Verdier =-=[49, 50]-=- in 2006 and in a simplified version in 2008, in which there is an incumbent politician who wants to be reelected, and a competitor. There are two periods with an election in the middle. The incumbent... |

87 |
Practical mathematical optimization: an introduction to basic optimization theory and classical and new gradient-based algorithms
- Snyman
- 2005
(Show Context)
Citation Context ... almost all branches of industry or in financial problems like strategic planning. There are several methods and algorithms to deal with optimization problems (for further information see for example =-=[52, 42]-=-). In this section we present some very famous conditions that will be applied later in the elaborate. Moreover we recall a well known theorem, which is also used in the following. 1.1.1 Karush-Kuhn-T... |

69 | Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria, Automatic Control
- Huang, Caines, et al.
(Show Context)
Citation Context ...oduction 4.1 An overview on mean field games Mean field games theory was developed in 2006 by Lasry and Lions [38, 39, 40], and independently in the engineering community by Caines, Huang and Malhamé =-=[9, 10]-=-, and it is a branch of game theory. It is therefore a set of concept and mathematical tools which help to model situations of infinitely many agents who take decisions in a context of strategic inter... |

69 |
Jeux à champ moyen. II. Horizon fini et contrôle optimal
- Lasry, Lions
(Show Context)
Citation Context ...and t ∈ T in which A(x) := {a ∈ A : x+ f(x, a) ∈ Q}. Chapter 4 Mean field games and oil production 4.1 An overview on mean field games Mean field games theory was developed in 2006 by Lasry and Lions =-=[38, 39, 40]-=-, and independently in the engineering community by Caines, Huang and Malhamé [9, 10], and it is a branch of game theory. It is therefore a set of concept and mathematical tools which help to model si... |

68 | Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle
- Huang, Malhamé, et al.
(Show Context)
Citation Context ...oduction 4.1 An overview on mean field games Mean field games theory was developed in 2006 by Lasry and Lions [38, 39, 40], and independently in the engineering community by Caines, Huang and Malhamé =-=[9, 10]-=-, and it is a branch of game theory. It is therefore a set of concept and mathematical tools which help to model situations of infinitely many agents who take decisions in a context of strategic inter... |

67 |
Resource Curse or Not: A Question of Appropriability.
- Boschini, Petterson, et al.
- 2007
(Show Context)
Citation Context ...that every Ri for i = 1, . . . , d is continuous and R ′ i, R ′′ i < 0. These assumptions on the derivatives mean that every Ri is a strictly decreasing and concave function of ei respectively. 5 See =-=[8]-=- for a discussion on the effects of the interaction between the type of resources that a country has and the quality of its institutions on the economic development. 6 An example of white elephant com... |

60 |
The implicit function theorem
- Krantz, Parks
- 2002
(Show Context)
Citation Context ... ( ∂(f1, . . . , fn) ∂(y1, . . . , yn) (x, φ(x)) )−1 ∂(f1, . . . , fn) ∂(x1, . . . , xl) (x, φ(x)) A proof of this result, along with further generalizations and examples, can be found for example in =-=[36]-=- and for the last part in [31]. Remark 1.4. The case n = 1 is of course the classic implicit function theorem for one dependent variable and one equation. 1.2 Optimal control theory Optimal control de... |

59 |
A numerical approach to the infinite horizon problem of deterministic control theory.
- FALCONE
- 1987
(Show Context)
Citation Context ...obi-Bellman equation related to an infinite horizon optimal control problem while in [25] Falcone and Giorgi deal with a finite horizon deterministic problem (for further works on this topic see also =-=[13, 22, 12, 24]-=-). 3.1 Model description We state briefly the new context of study and the new assumptions we make. Anything not explicitly mentioned are subject to the assumptions made in the previous chapter. In th... |

57 |
Mean field games and applications
- Lasry, Lions, et al.
- 2010
(Show Context)
Citation Context ...f the generic agent evolves deterministically according to an extraction rate and its target is to maximize an assigned income functional. In 2011 the same authors proposed a slightly different model =-=[30]-=- in which the reserve evolves according to a stochastic differential equation. In the following we present a procedure to rigorously derive the partial differential equation that describe the second m... |

51 |
Jeux à champ moyen. I. Le cas stationnaire
- Lasry, Lions
(Show Context)
Citation Context ...and t ∈ T in which A(x) := {a ∈ A : x+ f(x, a) ∈ Q}. Chapter 4 Mean field games and oil production 4.1 An overview on mean field games Mean field games theory was developed in 2006 by Lasry and Lions =-=[38, 39, 40]-=-, and independently in the engineering community by Caines, Huang and Malhamé [9, 10], and it is a branch of game theory. It is therefore a set of concept and mathematical tools which help to model si... |

50 |
Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations
- Ishii
- 1984
(Show Context)
Citation Context ...∈ R where Ω is an open domain of RN and the Hamiltonian F : R×R×RN → R is continuous. The term viscosity solutions first appear in a work of Crandall and Lions (see [18, 19] and, for further details, =-=[17, 33]-=-) in 1981, but the definition of solution had actually been given earlier by Evans [21] in 1980. Definition 1.6. A function u ∈ C is a viscosity subsolution of (1.7) if, for any ϕ ∈ C1, one has (1.8) ... |

44 |
Hamilton-Jacobi equations with state constraints
- Capuzzo-Dolchetta, Lions
- 1990
(Show Context)
Citation Context ... approach is a penalty argument. In optimization, two types of penalty function are commonly used: interior and exterior penalty functions. Considering an exterior penalty function, as for example in =-=[14]-=-, we can take a function ρ̃ ∈ BUC(R2) such that ρ̃ ≡ 0 in Q and for every ε > 0 there exists δ > 0 such that ρ̃(x) ≥ δ if dist(x,Q) ≥ ε. Then we can define the running cost ˜̀(x, a) := `(x, a) + 1 ε ρ... |

42 | White Elephant"
- Robinson, Torvik
- 2005
(Show Context)
Citation Context ...ndation in Palermo). So we choose to model patronage as offering job in the public sector to take the favors of voters. An alterative form of patronage can be the investment in white elephants, as in =-=[48]-=-, namely valuable but burdensome possessions or investment projects whose costs are out of proportion to its usefulness or worths 6 . In this framework the model under consideration concerns basically... |

36 | Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations
- Falcone, Ferretti
- 1994
(Show Context)
Citation Context ...obi-Bellman equation related to an infinite horizon optimal control problem while in [25] Falcone and Giorgi deal with a finite horizon deterministic problem (for further works on this topic see also =-=[13, 22, 12, 24]-=-). 3.1 Model description We state briefly the new context of study and the new assumptions we make. Anything not explicitly mentioned are subject to the assumptions made in the previous chapter. In th... |

34 |
Approximate solutions of the Bellman equation of deterministic control theory.
- I, ISHII
- 1984
(Show Context)
Citation Context ...obi-Bellman equation related to an infinite horizon optimal control problem while in [25] Falcone and Giorgi deal with a finite horizon deterministic problem (for further works on this topic see also =-=[13, 22, 12, 24]-=-). 3.1 Model description We state briefly the new context of study and the new assumptions we make. Anything not explicitly mentioned are subject to the assumptions made in the previous chapter. In th... |

28 |
On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming.
- Dolcetta
- 1983
(Show Context)
Citation Context ... a numerical point of view. Bellman equation can in fact be solved by induction numerically on a computer. There is a large literature about numerical approximations of the equation, in particular in =-=[11]-=- Capuzzo Dolcetta proposed an uniformly convergent approximation of the HamiltonJacobi-Bellman equation related to an infinite horizon optimal control problem while in [25] Falcone and Giorgi deal wit... |

28 |
Conditions d’unicité pour les solutions généralises des équations d’Hamilton-Jacobi du premier ordre
- Crandall, Lions
- 1981
(Show Context)
Citation Context ...ion (1.7) F (x, u(x), Du(x)) = 0 x ∈ R where Ω is an open domain of RN and the Hamiltonian F : R×R×RN → R is continuous. The term viscosity solutions first appear in a work of Crandall and Lions (see =-=[18, 19]-=- and, for further details, [17, 33]) in 1981, but the definition of solution had actually been given earlier by Evans [21] in 1980. Definition 1.6. A function u ∈ C is a viscosity subsolution of (1.7)... |

28 |
Continuity, chance and change: the character of the industrial revolution in Britain, Cambridge:
- WRIGLEY
- 1988
(Show Context)
Citation Context ...ce curse: a nonevolutive model 2.1 An overview on resource curse Scholars and economic historians traditionally emphasized the great benefits that natural resources give to a country (see for example =-=[56]-=- on the British case). However it seems that in many cases an abundance of natural resources leads to a poor economic development. In this regard in 1993 Auty [3] coined the term resource curse thesis... |

27 |
Politics of Patronage
- Tangri
- 1999
(Show Context)
Citation Context ...Development Corporation of Zambia, that was subject to a series of political and most importantly uneconomic directives on specific operational issues, including type and location of investments (see =-=[53]-=- for further details). Other examples can be founded in [35]. 7 We consider an open market, in which all economic actors have an equal opportunity of entry in that market. 2.4 Case d = 1: a single nat... |

26 |
Development Economics in Action: A Study of Economic Policies in Ghana. London:
- Killick
- 1978
(Show Context)
Citation Context ...ies of political and most importantly uneconomic directives on specific operational issues, including type and location of investments (see [53] for further details). Other examples can be founded in =-=[35]-=-. 7 We consider an open market, in which all economic actors have an equal opportunity of entry in that market. 2.4 Case d = 1: a single natural resource 16 This models the fact that, obviously, more ... |

21 |
Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation
- Allaire
- 2007
(Show Context)
Citation Context ...d) or, in the general case, the gradients of the active inequality constraints and the gradients of the equality constraints are linearly independent at x (for a detailed discussione see for instance =-=[1]-=-). Remark 1.2. The well known first order condition, that says that in an internal maximum (or minimum) point x it results ∇f(x) = 0, are a particular case of KKT conditions when we consider a solutio... |

17 |
Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications.
- Grasman, Herwaarden
- 1999
(Show Context)
Citation Context ...ay, that the density function m(t, R) of the reserves is transported by the optimal extraction rate according to a Kolmogorov-Fokker-Plank equation. The proof basically follows the ideas contained in =-=[28]-=-. Proposition 4.3. The density function m(t, R) ∈ C1,2 satisfies the KolmogorovFokker-Planck equation ∂tm+ ∂R(−qm) = ν 2 2 ∂2RR(R 2m) Proof. Let f ∈ C2 and R(t) be the solution of the stochastic diffe... |

17 | Why Do Some Resource-abundant Countries Succeed While Others Do Not?”
- TORVIK
- 2009
(Show Context)
Citation Context ...ublic employment. Within the framework of a nation with natural resources that are publicly owned such as oil, gas, diamond and other 4 For a general framework about the resource curse phenomenon see =-=[54]-=-. 2.3 Model description 15 minerals 5 and whose revenue from sales goes to the government, this model features an incumbent politician wishing to be reelected. There are two periods with the election ... |

15 |
Discrete dynamic programming and viscosity solutions of the Bellman equation
- Capuzzo-Dolcetta, Falcone
- 1989
(Show Context)
Citation Context |

13 |
On solving certain nonlinear partial differential equations by accretive operator methods.
- Evans
- 1980
(Show Context)
Citation Context ...erm viscosity solutions first appear in a work of Crandall and Lions (see [18, 19] and, for further details, [17, 33]) in 1981, but the definition of solution had actually been given earlier by Evans =-=[21]-=- in 1980. Definition 1.6. A function u ∈ C is a viscosity subsolution of (1.7) if, for any ϕ ∈ C1, one has (1.8) F (x0, u(x0), Dϕ(x0)) ≤ 0 at any local maximum point x0 ∈ R of u− ϕ. Similarly, u ∈ C i... |

13 | An approximation scheme for evolutive Hamilton-Jacobi equations
- Falcone, Giorgi
- 1999
(Show Context)
Citation Context ... equation, in particular in [11] Capuzzo Dolcetta proposed an uniformly convergent approximation of the HamiltonJacobi-Bellman equation related to an infinite horizon optimal control problem while in =-=[25]-=- Falcone and Giorgi deal with a finite horizon deterministic problem (for further works on this topic see also [13, 22, 12, 24]). 3.1 Model description We state briefly the new context of study and th... |

12 |
Numerical solution of dynamic programming equations, Appendix A
- Falcone
(Show Context)
Citation Context ...ables that minimize the cost functional. In this chapter we will present a model problem to focus on the main concept of the optimal control theory. Most of the contents can be founded for example in =-=[47, 4, 23]-=-. 1.2.1 A model problem: the infinite horizon discounted regulator Consider for example a control system governed by the state equation (1.2) { y′(t) = f(y(t), α(t)) t > 0 y(0) = x Here the control α ... |

8 |
The mathematical theory
- Pontryagin, Boltyanskii, et al.
- 1962
(Show Context)
Citation Context ...ables that minimize the cost functional. In this chapter we will present a model problem to focus on the main concept of the optimal control theory. Most of the contents can be founded for example in =-=[47, 4, 23]-=-. 1.2.1 A model problem: the infinite horizon discounted regulator Consider for example a control system governed by the state equation (1.2) { y′(t) = f(y(t), α(t)) t > 0 y(0) = x Here the control α ... |

7 |
poverty in southern Italy : a tale of two cities. Cambridge [Cambridgeshire
- Patronage
(Show Context)
Citation Context ...ibute public jobs or special favors in exchange for electoral support. It is widely believed that public employing is, politically speaking, a very profitable way to distribute rents (see for example =-=[15]-=-, about the phenomenon of political recommendation in Palermo). So we choose to model patronage as offering job in the public sector to take the favors of voters. An alterative form of patronage can b... |

7 |
Environment, human development and economic growth.
- Constantini, Monni
- 2008
(Show Context)
Citation Context ...xample it was studied in [2] if the resource curse phenomenon is due to presidentialism or parliamentarism or in [43] if it is caused by grabber friendly or producer friendly institutions, or more in =-=[16]-=- it was analyzed the causal relationship between economic growth, human development and sustainability. Moreover (see for example [20] for a very brief introduction) it has been suggested that the poo... |

6 |
Auty, "Sustaining Development in Mineral Economies: The Resource Curse Thesis
- M
- 1993
(Show Context)
Citation Context ...s give to a country (see for example [56] on the British case). However it seems that in many cases an abundance of natural resources leads to a poor economic development. In this regard in 1993 Auty =-=[3]-=- coined the term resource curse thesis to characterize the phenomenon whereby some countries rich in natural resources had lower economic growth than nation without this huge availability 4 . A crucia... |

6 |
Mean field games and oil production
- Guéant, Lasry, et al.
(Show Context)
Citation Context ...austible resource, but it is interesting to examine this in greater depth in a competitive situation and to understand the dynamics of exhaustion of a scarce resource. In 2010 Guéant, Larsy and Lions =-=[29]-=- published the first model of mean field games applied to this very old and studied question of exploitation of natural resources in which they consider a large number of oil producers, that can be vi... |

2 | Aslaksen, Constitutions and the resource curse - Andersen, S - 2008 |

2 |
Nonlinear Optimization
- Ruszczy«ski
- 2006
(Show Context)
Citation Context ... is an internal point of E, and consequently λ = 0. We want to point out that if some of the functions are nondifferentiable, subdifferential versions of KKT conditions are available (see for example =-=[51]-=-). 1.1.2 General implicit function theorem The well known implicit function theorem is one of the most important, and one of the oldest, paradigms in modern mathematics. The form of the implicit theor... |

1 |
The wealth of nations
- Diamond
- 2004
(Show Context)
Citation Context ...by grabber friendly or producer friendly institutions, or more in [16] it was analyzed the causal relationship between economic growth, human development and sustainability. Moreover (see for example =-=[20]-=- for a very brief introduction) it has been suggested that the poor development of countries has geographical origins also, such as location or climate, as well as political ones. 2.2 The considered f... |

1 |
Derivatives and Integrals of Multivariable Functions
- Guzman
- 2003
(Show Context)
Citation Context ... . , yn) (x, φ(x)) )−1 ∂(f1, . . . , fn) ∂(x1, . . . , xl) (x, φ(x)) A proof of this result, along with further generalizations and examples, can be found for example in [36] and for the last part in =-=[31]-=-. Remark 1.4. The case n = 1 is of course the classic implicit function theorem for one dependent variable and one equation. 1.2 Optimal control theory Optimal control deals with the problem of findin... |

1 |
Optimization and Economic Analysis
- Luptá£ik, Mathematical
- 2010
(Show Context)
Citation Context ... almost all branches of industry or in financial problems like strategic planning. There are several methods and algorithms to deal with optimization problems (for further information see for example =-=[52, 42]-=-). In this section we present some very famous conditions that will be applied later in the elaborate. Moreover we recall a well known theorem, which is also used in the following. 1.1.1 Karush-Kuhn-T... |

1 |
Institutions and the resource curse, Econ
- Mehlum, Moene, et al.
- 2006
(Show Context)
Citation Context ...natural resources. In this context there is an ample literature about this topic, for example it was studied in [2] if the resource curse phenomenon is due to presidentialism or parliamentarism or in =-=[43]-=- if it is caused by grabber friendly or producer friendly institutions, or more in [16] it was analyzed the causal relationship between economic growth, human development and sustainability. Moreover ... |

1 | Political foundations of the resource curse: an alternative formulation, preprint, available at http://www.svt.ntnu. no/iso/ragnar.torvik/simple%20model.pdf
- Robinson, Torvik, et al.
(Show Context)
Citation Context ...nagement of resources, or weak, ineffectual, unstable or corrupt institutions. We start our work from a model, which is halfway between economy and politics, developed by Robinson, Torvik and Verdier =-=[49, 50]-=- in 2006 and in a simplified version in 2008, in which there is an incumbent politician who wants to be reelected, and a competitor. There are two periods with an election in the middle. The incumbent... |

1 |
political parties
- Weingrod, Patrons
- 1968
(Show Context)
Citation Context ...maximizes a given income. We consider an incumbent that distribute its rent as patronage to influence the outcome of the election. Patronage is to be understood in the definition given by Weingrod in =-=[55]-=- that is the way in which party politicians distribute public jobs or special favors in exchange for electoral support. It is widely believed that public employing is, politically speaking, a very pro... |