We consider a class of optimal control problems of stochastic delay differential equations (SDDE) that arise in connection with optimal advertising under uncertainty for the introduction of a new product to the market, generalizing classical work of Nerlove and Arrow (1962). In particular, we deal with controlled SDDE where the delay enters both the state and the control. Following ideas of Vinter and Kwong (1981) (which however hold only in the deterministic case), we reformulate the problem as an infinite dimensional stochastic control problem to which we associate, through the dynamic programming principle, a second order Hamilton-Jacobi-Bellman equation. We show a verification theorem and we exhibit some simple cases where such equation admits an explicit smooth solution, allowing us to construct optimal feedback controls.
We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing a classical model of Nerlove and Arrow. In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work of two of the authors, the optimal advertising model is formulated as an infinite dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assum…
Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems
This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov-Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel-Legendre inequality or orthogonal poly…
Recently, sufficient conditions of stability or instability for time-delay systems have been proven to be necessary. In this way, a remarkable necessary and sufficient condition has then been developed by Gomez et al. It is presented as a simple test of positive definiteness of a matrix issued from the Lyapunov matrix. In this paper, an extension of this result is presented. Without going into details, the uniform discretization of the state has been replaced by projections on the first Legendre polynomials. Like Gomez et al., based on convergence arguments, the necessity is obtained in finite…
In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a $C^{1,α}$-partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising).
In this note, we present some complementary results on the infinite horizon optimal control for linear time-delay systems. We formally establish some properties of the matrices arising in the Bellman functional, and we prove that no concentrated delay term is present in the optimal control law.
In this paper we consider a family of optimal control problems for economic models whose state variables are driven by Delay Differential Equations (DDE's). We consider two main examples: an AK model with vintage capital and an advertising model with delay effect. These problems are very difficult to treat for three main reasons: the presence of the DDE's, that makes them infinite dimensional; the presence of state constraints; the presence of delay in the control. Our main goal is to develop, at a first stage, the Dynamic Programming approach for this family of problems. The Dynamic Programmi…
We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $α$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $α_-$ and $α_+$, depending on these bounds, and prove that for $α< α_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are con…
This paper, which is the natural continuation of a previous paper by the same authors, 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. 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 dire…
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. We embed the problem in a suitable Hilbert space H 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 sol…
Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases
Laboratory of Pure and Applied Mathematics, University of Laghouat（拉格瓦特大学纯数学与应用数学实验室）；Department of Mathematics and Computer Science, Faculty of Science and Technology, University of Ghardaia（加达亚大学科学与技术学院数学与计算机科学系）
