Fausto Gozzi | LUISS Guido Carli (original) (raw)

Papers by Fausto Gozzi

arXiv (Cornell University), Jan 24, 2021

We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing co... more We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing constraints and a finite retirement date. Similarly to [7], wages evolve in a path-dependent way, but the presence of a finite retirement time leads to a novel, twostage infinite dimensional stochastic optimal control problem with explicit optimal controls in feedback form. This is possible as we find an explicit solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of parabolic type. The identification of the optimal feedbacks is delicate due to the presence of time-dependent state constraints, which appear to be new in the infinite dimensional stochastic control literature. The explicit solution allows us to study the properties of optimal strategies and discuss their implications for portfolio choice. As opposed to models with Markovian dynamics, path dependency can now modulate the hedging demand arising from the implicit holding of risky assets in human capital, leading to richer asset allocation predictions consistent with wage rigidity and the agents learning about their earning potential.

Science Advances, May 24, 2023

The global pandemic of COVID-19 has underlined the need for more coordinated responses to emergen... more The global pandemic of COVID-19 has underlined the need for more coordinated responses to emergent pathogens. These responses need to balance epidemic control in ways that concomitantly minimize hospitalizations and economic damages. We develop a hybrid economic-epidemiological modeling framework that allows us to examine the interaction between economic and health impacts over the first period of pathogen emergence when lockdown, testing, and isolation are the only means of containing the epidemic. This operational mathematical setting allows us to determine the optimal policy interventions under a variety of scenarios that might prevail in the first period of a large-scale epidemic outbreak. Combining testing with isolation emerges as a more effective policy than lockdowns, substantially reducing deaths and the number of infected hosts, at lower economic cost. If a lockdown is put in place early in the course of the epidemic, it always dominates the “laissez-faire” policy of doing nothing.

アクチュアリ-ジャ-ナル, Dec 1, 2010

Probability theory and stochastic modelling, 2017

This chapter is devoted to the presentation of the \(L^2\) theory for the existence and uniquenes... more This chapter is devoted to the presentation of the \(L^2\) theory for the existence and uniqueness of mild solutions for a class of second-order infinite-dimensional HJB equations in Hilbert spaces through a perturbation approach.

Probability theory and stochastic modelling, 2017

In this chapter we discuss the connection between the study of infinite-dimensional stochastic op... more In this chapter we discuss the connection between the study of infinite-dimensional stochastic optimal control problems and that of second-order Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces.

Siam Journal on Control and Optimization, 2011

This paper, which is the natural continuation of [14], studies a class of optimal control problem... more This paper, which is the natural continuation of [14], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In [14] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. The goal of the present paper is to exploit the regularity result of [14] to prove a Verification Theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of V and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-tobuild models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

RePEc: Research Papers in Economics, 2004

Endogenous growth, Optimal control with mixed constraints, von Neumann growth model,

RePEc: Research Papers in Economics, Nov 19, 2009

In this paper we give a sufficient and almost necessary condition for the existence of optimal st... more In this paper we give a sufficient and almost necessary condition for the existence of optimal strategies in linear multisector models when time is continuous and the the instantaneous utility function of the representative agent has two properties: (a) the intertemporal elasticity of substitution is constant over time and (b) preferences are concave and homothetic.

arXiv (Cornell University), May 12, 2009

This paper, which is the natural continuation of [14], studies a class of optimal control problem... more This paper, which is the natural continuation of [14], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In [14] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. The goal of the present paper is to exploit the regularity result of [14] to prove a Verification Theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of V and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-tobuild models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

arXiv (Cornell University), Jul 9, 2009

This paper, which is the natural continuation of [21], studies a class of optimal control problem... more This paper, which is the natural continuation of [21], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. This class includes some problems arising in economics, in particular the so-called models with time to build. In [21] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. Therein the main result is that the value function V solves the HJB equation and has continuous classical derivative in the direction of the "present". The goal of the present paper is to exploit such result to find optimal feedback strategies for the problem. While it is easy to define formally a feedback strategy in classical sense the proof of its existence and of its optimality is hard due to lack of full regularity of V and to the infinite dimension. Finally, we show some approximation results that allow us to apply our main theorem to obtain ε-optimal strategies for a wider class of problems.

HAL (Le Centre pour la Communication Scientifique Directe), 2017

Probability theory and stochastic modelling, 2017

Economic Theory

A large number of recent studies consider a compartmental SIR model to study optimal control poli... more A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton–Jacobi–Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.

arXiv (Cornell University), Jun 1, 2022

A large number of recent studies consider a compartmental SIR model to study optimal control poli... more A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.

Vintage capital growth models have been at the heart of growth theory in the 60s. This research l... more Vintage capital growth models have been at the heart of growth theory in the 60s. This research line collapsed in the late 60s with the so-called embodiment controversy and the technical sophisitication of the vintage models. This paper analyzes the astonishing revival of this literature in the 90s. In particular, it outlines three methodological breakthroughs explaining this resurgence: a growth accounting revolution, taking advantage of the availability of new time series, an optimal control revolution allowing to safely study vintage capital optimal growth models, and a vintage human capital revolution, along with the rise of economic demography, accounting for the vintage structure of human capital similarly to physical capital age structuring. The related literature is surveyed.

... Peter Groenewegen Marshall's Evolutionary Economics Tizi... more ... Peter Groenewegen Marshall's Evolutionary Economics Tiziano Raffaelli Money, Time and Rationality in Max Weber Austrian Connections Stephen D. Parsons ... Zouache 71 Consumption as an Investment The fear of goods from Hesiod to Adam Smith Cosimo Perrotta 72 Jean ...

SSRN Electronic Journal, 2021

In this paper, we investigate how a transitory lockdown of a sector of the economy may have chang... more In this paper, we investigate how a transitory lockdown of a sector of the economy may have changed our habits and, therefore, altered the goods' demand permanently. In a two-sector infinite horizon economy, we show that the demand of the goods produced by the sector closed during the lockdown could shrink or expand with respect to their pre-pandemic level depending on the lockdown's duration and the habits' strength. We also show that the end of a lockdown may be characterized by a price surge due to a combination of strong demand of both goods and rigidities in production.

Journal of Optimization Theory and Applications

In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bel... more In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

Stochastic Optimal Control in Infinite Dimension, 2017

We recall some basic notions of measure theory and give a short introduction to random variables ... more We recall some basic notions of measure theory and give a short introduction to random variables and the theory of the Bochner integral.

arXiv (Cornell University), Jan 24, 2021

We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing co... more We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing constraints and a finite retirement date. Similarly to [7], wages evolve in a path-dependent way, but the presence of a finite retirement time leads to a novel, twostage infinite dimensional stochastic optimal control problem with explicit optimal controls in feedback form. This is possible as we find an explicit solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of parabolic type. The identification of the optimal feedbacks is delicate due to the presence of time-dependent state constraints, which appear to be new in the infinite dimensional stochastic control literature. The explicit solution allows us to study the properties of optimal strategies and discuss their implications for portfolio choice. As opposed to models with Markovian dynamics, path dependency can now modulate the hedging demand arising from the implicit holding of risky assets in human capital, leading to richer asset allocation predictions consistent with wage rigidity and the agents learning about their earning potential.

Science Advances, May 24, 2023

The global pandemic of COVID-19 has underlined the need for more coordinated responses to emergen... more The global pandemic of COVID-19 has underlined the need for more coordinated responses to emergent pathogens. These responses need to balance epidemic control in ways that concomitantly minimize hospitalizations and economic damages. We develop a hybrid economic-epidemiological modeling framework that allows us to examine the interaction between economic and health impacts over the first period of pathogen emergence when lockdown, testing, and isolation are the only means of containing the epidemic. This operational mathematical setting allows us to determine the optimal policy interventions under a variety of scenarios that might prevail in the first period of a large-scale epidemic outbreak. Combining testing with isolation emerges as a more effective policy than lockdowns, substantially reducing deaths and the number of infected hosts, at lower economic cost. If a lockdown is put in place early in the course of the epidemic, it always dominates the “laissez-faire” policy of doing nothing.

アクチュアリ-ジャ-ナル, Dec 1, 2010

Probability theory and stochastic modelling, 2017

This chapter is devoted to the presentation of the \(L^2\) theory for the existence and uniquenes... more This chapter is devoted to the presentation of the \(L^2\) theory for the existence and uniqueness of mild solutions for a class of second-order infinite-dimensional HJB equations in Hilbert spaces through a perturbation approach.

Probability theory and stochastic modelling, 2017

In this chapter we discuss the connection between the study of infinite-dimensional stochastic op... more In this chapter we discuss the connection between the study of infinite-dimensional stochastic optimal control problems and that of second-order Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces.

Siam Journal on Control and Optimization, 2011

This paper, which is the natural continuation of [14], studies a class of optimal control problem... more This paper, which is the natural continuation of [14], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In [14] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. The goal of the present paper is to exploit the regularity result of [14] to prove a Verification Theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of V and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-tobuild models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

RePEc: Research Papers in Economics, 2004

Endogenous growth, Optimal control with mixed constraints, von Neumann growth model,

RePEc: Research Papers in Economics, Nov 19, 2009

In this paper we give a sufficient and almost necessary condition for the existence of optimal st... more In this paper we give a sufficient and almost necessary condition for the existence of optimal strategies in linear multisector models when time is continuous and the the instantaneous utility function of the representative agent has two properties: (a) the intertemporal elasticity of substitution is constant over time and (b) preferences are concave and homothetic.

arXiv (Cornell University), May 12, 2009

This paper, which is the natural continuation of [14], studies a class of optimal control problem... more This paper, which is the natural continuation of [14], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In [14] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. The goal of the present paper is to exploit the regularity result of [14] to prove a Verification Theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of V and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-tobuild models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

arXiv (Cornell University), Jul 9, 2009

This paper, which is the natural continuation of [21], studies a class of optimal control problem... more This paper, which is the natural continuation of [21], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. This class includes some problems arising in economics, in particular the so-called models with time to build. In [21] the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. Therein the main result is that the value function V solves the HJB equation and has continuous classical derivative in the direction of the "present". The goal of the present paper is to exploit such result to find optimal feedback strategies for the problem. While it is easy to define formally a feedback strategy in classical sense the proof of its existence and of its optimality is hard due to lack of full regularity of V and to the infinite dimension. Finally, we show some approximation results that allow us to apply our main theorem to obtain ε-optimal strategies for a wider class of problems.

HAL (Le Centre pour la Communication Scientifique Directe), 2017

Probability theory and stochastic modelling, 2017

Economic Theory

A large number of recent studies consider a compartmental SIR model to study optimal control poli... more A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton–Jacobi–Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.

arXiv (Cornell University), Jun 1, 2022

A large number of recent studies consider a compartmental SIR model to study optimal control poli... more A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.

Vintage capital growth models have been at the heart of growth theory in the 60s. This research l... more Vintage capital growth models have been at the heart of growth theory in the 60s. This research line collapsed in the late 60s with the so-called embodiment controversy and the technical sophisitication of the vintage models. This paper analyzes the astonishing revival of this literature in the 90s. In particular, it outlines three methodological breakthroughs explaining this resurgence: a growth accounting revolution, taking advantage of the availability of new time series, an optimal control revolution allowing to safely study vintage capital optimal growth models, and a vintage human capital revolution, along with the rise of economic demography, accounting for the vintage structure of human capital similarly to physical capital age structuring. The related literature is surveyed.

... Peter Groenewegen Marshall's Evolutionary Economics Tizi... more ... Peter Groenewegen Marshall's Evolutionary Economics Tiziano Raffaelli Money, Time and Rationality in Max Weber Austrian Connections Stephen D. Parsons ... Zouache 71 Consumption as an Investment The fear of goods from Hesiod to Adam Smith Cosimo Perrotta 72 Jean ...

SSRN Electronic Journal, 2021

In this paper, we investigate how a transitory lockdown of a sector of the economy may have chang... more In this paper, we investigate how a transitory lockdown of a sector of the economy may have changed our habits and, therefore, altered the goods' demand permanently. In a two-sector infinite horizon economy, we show that the demand of the goods produced by the sector closed during the lockdown could shrink or expand with respect to their pre-pandemic level depending on the lockdown's duration and the habits' strength. We also show that the end of a lockdown may be characterized by a price surge due to a combination of strong demand of both goods and rigidities in production.

Journal of Optimization Theory and Applications

In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bel... more In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

Stochastic Optimal Control in Infinite Dimension, 2017

We recall some basic notions of measure theory and give a short introduction to random variables ... more We recall some basic notions of measure theory and give a short introduction to random variables and the theory of the Bochner integral.