Finite-Dimensional Representations for Controlled Diffusions with Delay

Federico, Salvatore; Tankov, Peter

doi:10.1007/s00245-014-9256-2

Cited by 14 publications

(9 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The new state equation is the formal counterpart of the SDDE of the original problem, and essentially an infinite-dimensional SDE in a suitable Hilbert space or Banach space, in which the equivalent infinite-dimensional problem lives. See also [17] for sufficient conditions under which the infinite-dimensional setting of controlled SDDEs admits exact and approximate finite-dimensional representations. Then one should show the equivalence between the original problem with delay and the infinite-dimensional problem without delay in the mild solution sense.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control for Stochastic Delay Evolution Equations

Meng

Shen

2015

Appl Math Optim

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal Control for Stochastic Delay Evolution Equations

Meng

Shen

2015

Appl Math Optim

View full text Add to dashboard Cite

show abstract

“…To get this goal one can proceed as in [29] by assuming more regularity on the data of the problem. More precisely, assuming that l 0 ∈ C 1 b (R) and that the Hamiltonian 17 It must be noted that, under suitable restrictions on the data, one can treat (stochastic) optimal control problems with delay avoiding to look at them as infinite dimensional systems (see [18]). However, this is possible only in very special cases, leaving out a lot of of concrete applications.…”

Section: Stochastic Optimal Control With Delay In the Control Variablementioning

confidence: 99%

Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

Federico¹,

Gozzi²

2018

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein-Uhlenbeck processes, when the associated Hamilton-Jacobi-Bellman (HJB) equation admits a mild solution (in the sense of [16]). The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature (see e.g. [26]), that the mild solution is a strong solution, i.e. a suitable limit of classical solutions of the HJB equation. To achieve the goal we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.

show abstract

“…A typical approach to deal with optimal control problem with delay is to represent the control system in an appropriate infinite dimensional space without delay. For instance, under the sufficient condition of [12], we derive the equivalence between the original finite dimensional problem with delay and the infinite dimensional problem without delay in the sense of weak solution, then this problem can be solved by discussing the corresponding infinite dimensional HJB equations. For Pontryagin's maximum principle, the main idea is to use the way of variation to solve the extremum problem of dynamic system.…”

Section: Introductionmentioning

confidence: 99%

Infinite Horizon Stochastic Delay Evolution Equations in Hilbert Spaces and Stochastic Maximum Principle

Zhou

Dai

et al. 2022

Taiwanese J. Math.

View full text Add to dashboard Cite

In this paper, a class of infinite horizon optimal control problems is established, where the state equation is given by a stochastic delay evolution equation (SDEE), and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). Firstly, we extend the form of Itô formula. After that, we establish the priori estimate for the solution to ABSEEs, and then the existence and uniqueness results of ABSEEs on infinite horizon are obtained. Finally, we establish necessary and sufficient conditions of stochastic maximum principle for infinite horizon optimal control problem in the form of Pontryagin's maximum principle.

show abstract

Finite-Dimensional Representations for Controlled Diffusions with Delay

Cited by 14 publications

References 33 publications

Optimal Control for Stochastic Delay Evolution Equations

Optimal Control for Stochastic Delay Evolution Equations

Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

Infinite Horizon Stochastic Delay Evolution Equations in Hilbert Spaces and Stochastic Maximum Principle

Contact Info

Product

Resources

About