Fausto Gozzi

Pension Funds with a Minimum Guarantee: A Stochastic Control Approach

In this paper we propose and study a continuous-time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits, and the level of wealth is constrained to stay above a “solvency level.” The fund manager can invest in a riskless asset and in a risky asset, but borrowing and short selling are prohibited. We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level.The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton–Jacobi–Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution.

Technology adoption and accumulation in a vintage-capital model

We present a model of capital accumulation and technology adoption in a vintage-capital framework. The model is an infinite-horizon/infinite-dimensional optimal control model: the firm employs a continuum of technologies (a continuum of heterogeneous capital goods). Capital goods are technology specific, their technology is related to vintage and technology progress. The entrepreneur maximizes the profits obtained by employing a continuum of technologies under the assumption of constant returns to scale and bearing adjustment costs for gross investments. The diffusion of a new technology is established by allowing the entrepreneur to invest in vintage capital goods.

Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control

The paper is concerned with fully nonlinear second order Hamilton--Jacobi--Bellman--Isaacs equations of elliptic type in separable Hilbert spaces which have unbounded first and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved. A stochastic optimal control problem driven by a parabolic stochastic PDE with control of Dirichlet type on the boundary is considered. It is proved that the value function of this problem is the unique viscosity solution of the associated Hamilton--Jacobi--Bellman equation.

Regularity of the Minimum Time Function and Minimum Energy Problems: The Linear Case

We prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable Hilbert space H. Part of the results are new also in the finite dimensional case. The regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy.

Erratum: A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

We correct the proof of Theorem 4.1 in Gozzi, Swiech, and Zhou [SIAM J. Control Optim., 43 (2005), pp. 2009-2019] by imposing additional conditions on the viscosity subsolution

HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, I: Regularity of Viscosity Solutions

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ON THE DYNAMIC PROGRAMMING APPROACH FOR OPTIMAL CONTROL PROBLEMS OF PDE'S WITH AGE STRUCTURE

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A boundary-value problem for Hamilton-Jacobi equations in Hilbert spaces

We study 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; see [P. K. Asea and P. J. Zak, J. Econom. Dynam. Control, 23 (1999), pp. 1155-1175; M. Bambi, J. Econom. Dynam. Control, 32 (2008), pp. 1015-1040; F. E. Kydland and E. C. Prescott, Econometrica, 50 (1982), pp. 1345-1370]. We embed the problem in a suitable Hilbert space

Stochastic optimal control in infinite dimension : dynamic programming and HJB equations

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INVESTMENT/CONSUMPTION PROBLEM IN ILLIQUID MARKETS WITH REGIME-SWITCHING ∗

and consider the associated Hamilton-Jacobi-Bellman (HJB) equation. This kind of infinite dimensional HJB equation has not been previously studied and is difficult due to the presence of state constraints and the lack of smoothing properties of the state equation. Our main result on the regularity of solutions to such an HJB equation seems to be entirely new. More precisely, we prove that the value function is continuous in a sufficiently big open set of