Necessary conditions for optimality in relaxed stochastic control problems

Mezerdi, Brahim; Bahlali, Seid

doi:10.1080/1045112021000025925

Cited by 18 publications

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“…The family of relaxed controls ((µ n ) n≥0 , µ) is tight in R the space of probability measures on [0; T ] × A Proof. see Mezerdi, and Bahlali (2002) Lemma 3. The family of martingale measures (( N n ) n≥0 , N µ ) is tight in the space B. G. Hanane and B. Mezerdi, Afrika Statistika, Vol.…”

Section: Approximation Of Trajectoriesmentioning

confidence: 99%

“…These are called relaxed controls. For more details see Mezerdi, and Bahlali (2002); Mezerdi and Bahlali (2000) . The problem now is to define rigorously the dynamics associated to a relaxed control.…”

Section: The Relaxed Control Problemmentioning

confidence: 99%

“…The first existence result of an optimal relaxed control is proved by Fleming (1977), for the SDE's with uncontrolled diffusion coefficient and no jump term. For such systems of SDE's a maximum principle has been established in Bahlali et al (2007Bahlali et al ( , 2006 ;Mezerdi, and Bahlali (2002). For mean-field systems one can refer to Bahlali et al (2014Bahlali et al ( , 2017a.…”

Section: Introductionmentioning

confidence: 99%

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The relaxed stochastic maximum principle in optimal control of diffusions with controlled jumps

Gherbal¹,

Mezerdi²

2017

AJAS

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Abstract. This paper is concerned with optimal control of systems driven by stochastic differential equations (SDEs), with jump processes, where the control variable appears in the drift and in the jump term. We study the relaxed problem, in which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by a counting measure valued process called relaxed Poisson measure such that the compensator is a product measure. Under some conditions on the coefficients, we prove that every diffusion process associated to a relaxed control is a limit of a sequence of diffusion processes associated to strict controls. As a consequence, we show that the strict and the relaxed control problems have the same value function. Using similar arguments, we prove the existence of an optimal relaxed control. In a second step, we establish a maximum principle for this type of relaxed problem.Key words: Stochastic control, Stochastic differential equation, jump process, optimal control, relaxed control -maximum principle. AMS 2010 Mathematics Subject Classification : 93E20, 60H10 Résumé (French Abstract). L'objectif de cet article est l'étude du contrôle optimal de systèmes dirigés par deséquations différentielles stochastiques (EDS), présentant des sauts, où le paramètre de contrôle apparaît aussi bien dans le drift que dans le terme de saut. Nousétudions le problème relaxé, dans lequel les contrôles admissibles sont des processusà valeurs mesures et la variable d'état est gouvernée par une EDS dirigée par une mesure de comptage appelée mesure de Poisson relaxée, dont le compensateur est une mesure produit. Sous certaines hypothèses sur les coefficients, nous montrons que tout processus de diffusion * Corresponding author Brahim Mezerdi : bmezerdi@yahoo.fr Ben Gherbal Hanane : hananebengherbal@yahoo.fr B. G. Hanane and B. Mezerdi, Afrika Statistika, Vol. 12 (2), 2017, 1287 -1312. The relaxed stochastic maximum principle in optimal control of diffusions with controlled jumps. 1288associéà un contrôle relaxé est limite d'une suite de diffusions associéesà des contrôles stricts. Comme conséquence, nousétablissons que les problèmes de contrôle strict et relaxé ont la même fonction de valeurs. En utilisant des arguments similaires, nous montrons l'existence d'un contrôle optimal relaxé. Dans une deuxièmeétape, nous démontrons un principe du maximum pour ce type de problème relaxé.

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Section: Approximation Of Trajectoriesmentioning

confidence: 99%

Section: The Relaxed Control Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The relaxed stochastic maximum principle in optimal control of diffusions with controlled jumps

Gherbal¹,

Mezerdi²

2017

AJAS

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“…The first existence of an optimal relaxed control has been proved in Fleming (1977), for classical Ito SDEs, where only the drift is controlled. The case of an SDE where the diffusion coefficient depends explicitly on the control variable has been solved in El Karoui et al (1987);Haussmann (1986), where the optimal relaxed control is shown to be Markovian, see also Haussmann and Lepeltier (1990); Mezerdi and Bahlali (2002); Bahlali et al (2006). Existence results for systems driven by backward and forward-backward SDEs have been investigated in Bahlali et al (2010Bahlali et al ( , 2011Buckdahn et al (2010).…”

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confidence: 99%

Existence of optimal controls for systems governed by mean-field stochastic differential equations

Bahlali¹,

Mezerdi

2014

jas

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Abstract. In this paper, we study the existence of an optimal control for systems, governed by stochastic differential equations of mean-field type. For non linear systems, we prove the existence of an optimal relaxed control, by using tightness techniques and Skorokhod selection theorem. The optimal control is a measure valued process defined on another probability space. In the case where the coefficients are linear maps and the cost functions are convex, we prove by using weak convergence techniques, the existence of an optimal strict control, adapted to the initial filtration.Résumé. Dans cet article on s'intéresseà l'existence d'un contrôle optimal, pour des systèmes gouvernés par deséquations différentielles stochastiques de type champ moyen. Pour les systèmes non linéaires, on démontre un résultat d'existence d'un contrôle optimal relaxé, en utilisant des techniques de tension et le théorème de sélection de Skorokhod. Le contrôle optimal obtenu est un processusà valeurs mesures, défini sur un autre espace de probabilité. Dans le cas où les coefficients sont linéaires et les fonctions de coût sont convexes, on démontre en utilisant des techniques de convergence faible, l'existence d'un contrôle optimal strict, adaptéà la filtration initiale.

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“…The first result in this direction has been established in Mezerdi and Bahlali (2002), where a stochastic maximum principle for relaxed controls, in the case of uncontrolled diffusion coefficient has been given by using the first order adjoint process (see also Bahlali et al, 2007 the extension to singular control problems). The case of S. Labed and B. Mezerdi, Afrika Statistika, Vol.…”

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confidence: 99%

The maximum principle in optimal control of systems driven by martingale measures

Labed¹,

Mezerdi²

2017

JAS

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Abstract. We study the relaxed optimal stochastic control problem for systems governed by stochastic differential equations (SDEs), driven by an orthogonal continuous martingale measure, where the control is allowed to enter both the drift and diffusion coefficient. The set of admissible controls is a set of measure-valued processes. Necessary conditions for optimality for these systems in the form of a maximum principle are established by means of spike variation techniques. Our result extends Peng's maximum principle to the class of measure valued controls.Résumé. Nousétudions les problèmes de contrôle stochastique relaxés pour des systèmes gouvernés par deséquations différentielles stochastiques (EDSs), dirigées par des mesures martingales orthogonales continues, avec un drift et un coefficient de diffusion contrôlé. L'ensemble des contrôles admissibles est constitué de processusà valeurs mesures. Onétablit des conditions nécessaires d'optimalité en utilisant des preturbations fortes. Notre résultat généralise le principe du maximum de Peng pour la classe de contrôlesà valeurs mesures.

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Necessary conditions for optimality in relaxed stochastic control problems

Cited by 18 publications

References 19 publications

The relaxed stochastic maximum principle in optimal control of diffusions with controlled jumps

The relaxed stochastic maximum principle in optimal control of diffusions with controlled jumps

Existence of optimal controls for systems governed by mean-field stochastic differential equations

The maximum principle in optimal control of systems driven by martingale measures

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