A Maximum Principle for Infinite Horizon Delay Equations

Agram, Nacira; Haadem, Sven; Øksendal, Bernt; Proske, Frank

doi:10.1137/120882809

Cited by 56 publications

(50 citation statements)

References 17 publications

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“…One direction involves a system of three-coupled adjoint equations, with two BSDEs and one backward ordinary differential equation (ODE). See, for example [1,6,28]. Different from the previous works [33] proposed a system of three-coupled BSDE as adjoint equations for a stochastic system with delay and mean-field terms.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control for Stochastic Delay Evolution Equations

Meng

Shen

2015

Appl Math Optim

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In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linearquadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

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Section: Introductionmentioning

confidence: 99%

Optimal Control for Stochastic Delay Evolution Equations

Meng

Shen

2015

Appl Math Optim

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“…Although it is possible to study the problem using a modified version of the Pontryagin Maximum Principle (PMP) (see, e.g., Agram et al [6]), this approach hardly allows the identification of an explicit formulation of the optimal policy (as we do) because of the mixed type equation resulting from the PMP in the presence of retarded control.…”

Section: Manuscriptmentioning

confidence: 99%

“…We now look at the properties of the lower bound (for admissible controls) c m (·), which solves equation (6). The characteristic equation associated with the delay equation (6) (e.g.…”

Section: Is Immediate Using (8) and Formula (4) ⊓ ⊔mentioning

confidence: 99%

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Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

Augeraud-Véron

Bambi

Gozzi

2017

J Optim Theory Appl

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Abstract:In this paper, we study an economic model where internal habits play a role. Their formation is described by a more general functional form than is usually assumed in the literature, because a finite memory effect is allowed. Indeed, the problem becomes the optimal control of a standard ordinary differential equation, with the past of the control entering both the objective function and an inequality constraint. Therefore, the problem is intrinsically infinite dimensional. To solve this model, we apply the dynamic programming approach and we find an explicit solution for the associated Hamilton-Jacobi-Bellman equation, which lets us write the optimal strategies in feedback form. Therefore, we contribute to the existing literature in two ways. Firstly, we fully develop the dynamic programming approach to a type of problem not studied in previous contributions. Secondly, we use this result to unveil the global dynamics of an economy characterized by generic internal habits. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2 Emmanuelle Augeraud-Veron et al.differential equation, with the past of the control entering both the objective function and an inequality constraint. Therefore, the problem is intrinsically infinite dimensional. To solve this model, we apply the dynamic programming approach and we find an explicit solution for the associated Hamilton-JacobiBellman equation, which lets us write the optimal strategies in feedback form.Therefore, we contribute to the existing literature in two ways. Firstly, we fully develop the dynamic programming approach to a type of problem not studied in previous contributions. Secondly, we use this result to unveil the global dynamics of an economy characterized by generic internal habits.

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“…Therefore, it is of great relevance to develop the maximum principles under control systems with delay and apply these maximum principles to solve practical problems arising in the real world. See, for example, [16,17,[19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach

Meng

Shen

2015

Journal of Computational and Applied Mathematics

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a b s t r a c tThis paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay. Firstly, some existence and uniqueness results are proved for a jumpdiffusion mean-field stochastic delay differential equation and a jump-diffusion meanfield advanced backward stochastic differential equation. Then necessary and sufficient maximum principles for control systems of mean-field type and with delay are established under certain conditions. A mean-field, delayed, linear-quadratic control problem is finally discussed using the obtained maximum principles.

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A Maximum Principle for Infinite Horizon Delay Equations

Cited by 56 publications

References 17 publications

Optimal Control for Stochastic Delay Evolution Equations

Optimal Control for Stochastic Delay Evolution Equations

Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach

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