A stochastic maximum principle in mean-field optimal control problems for jump diffusions

Chighoub, Farid; Mezerdi, Brahim

doi:10.1016/j.ajmsc.2013.02.001

Cited by 10 publications

(7 citation statements)

References 16 publications

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“…where, H θ is the risk-sensitive Hamiltonian given by (6). Using the notation (4), we have the following relation between the drift termα 1 in (37) and the gradient of the risk-sensitive Hamiltonian H θ :…”

Section: Transformation Of the Hamiltonianmentioning

confidence: 99%

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A stochastic maximum principle for risk-sensitive mean-field-type control

Djehiche

Tembiné

Tempone

2014

53rd IEEE Conference on Decision and Control

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In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the value-function imposed in Lim and Zhou (2005) need not to be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.Index Terms. time inconsistent stochastic control, maximum principle, meanfield SDE, risk-sensitive control, logarithmic transformation.Abbreviated title. Risk-sensitive control of SDEs of mean field type AMS subject classification. 93E20, 60H30, 60H10, 91B28.

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Section: Transformation Of the Hamiltonianmentioning

confidence: 99%

“…We also introduce the risk-sensitive Hamiltonian: for θ ∈ lR and (p, q, ℓ) ∈ lR × lR × lR, H θ (t, X, u, p, q, ℓ) := b(t, X, E [X] , u)p + σ(t, X, E [X] , u)(q + θℓp) − f (t, X, E [X] , u). (6) We have H = H 0 .…”

Section: Introductionmentioning

confidence: 99%

A stochastic maximum principle for risk-sensitive mean-field-type control

Djehiche

Tembiné

Tempone

2014

53rd IEEE Conference on Decision and Control

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“…The main drawback, when dealing with such mean-field stochastic control problems, is that the Bellmann principle of optimality does not hold. For this kind of problems, the stochastic maximum principle, provides a powerful tool to solve them, see [1,6,7,9,10,18,27]. One can refer also to the recent book [2] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the relaxed mean-field stochastic control problem

Bahlali

Mezerdi

2018

Stoch. Dyn.

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This paper is concerned with optimal control problems for systems governed by mean-field stochastic differential equation, in which the control enters both the drift and the diffusion coefficient. We prove that the relaxed state process, associated with measure valued controls, is governed by an orthogonal martingale measure rather that a Brownian motion. In particular, we show by a counter example that replacing the drift and diffusion coefficient by their relaxed counterparts does not define a true relaxed control problem. We establish the existence of an optimal relaxed control, which can be approximated by a sequence of strict controls. Moreover under some convexity conditions, we show that the optimal control is realized by a strict control.

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“…Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2015.2406973, IEEE Transactions on Automatic Control 6 We consider the following transform…”

Section: B Transformation Of the First Order Adjoint Process And Promentioning

confidence: 99%

A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

Djehiche

Tembiné

Tempone

2015

IEEE Trans. Automat. Contr.

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In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the valuefunction imposed in Lim and Zhou (2005) needs not be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.

show abstract

A stochastic maximum principle in mean-field optimal control problems for jump diffusions

Cited by 10 publications

References 16 publications

A stochastic maximum principle for risk-sensitive mean-field-type control

A stochastic maximum principle for risk-sensitive mean-field-type control

On the relaxed mean-field stochastic control problem

A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

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