A stochastic maximum principle for processes driven by <i>G</i>‐Brownian motion and applications to finance

Sun, Zhongyang; Zhang, Xin; Guo, Junyi

doi:10.1002/oca.2299

Cited by 7 publications

(4 citation statements)

References 36 publications

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“…Let us introduce the adjoint process, which is a G-backward stochastic differential equation (G-BSDE in short). We proceed as in [8], [49].and [7].…”

Section: The Maximum Principle For Strict Controlmentioning

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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“…Let us introduce the adjoint process, which is a G-backward stochastic differential equation (G-BSDE in short). We proceed as in [8], [49].and [7].…”

Section: The Maximum Principle For Strict Controlmentioning

confidence: 99%

“…Finally, based on the remark 5.2 in [49] if we assume that in equation ( 21) k = 0 q.s and we define the Hamiltonian H from [0; T ]×R n ×A×R n ×R n×m ×L 2 m into R by H(t, x, u, p, q, r(.)) = h(t, x t , u t ) + pb(t, x t , u t ) +qσ(t, x t ) + Γ r t (θ)f (s, x t , θ, u t )υ(dθ).…”

Section: The Maximum Principle For Near Optimal Controlsmentioning

confidence: 99%

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

Gherbal¹,

Mezerdi²

2017

AJAS

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“…where coefficients , and are defined in the following section. In a departure from classical martingale representation, the G-martingale is bifurcated into two segments: the symmetric Gmartingale , where also qualifies as a G-martingale, and the decreasing Gmartingale component , which, although absent from classical theory, is integral to this novel framework (refer to [26,29,30]). For a comprehensive understanding, readers are directed to the work of Peng [20,22,25,27], among others.…”

Section: Introductionmentioning

confidence: 99%

<i>G</i>-stochastic maximum principle for risk-sensitive control problem and its applications

Dassa,

Chala

2023

PUQR

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This study advances the G-stochastic maximum principle (G-SMP) from a riskneutral framework to a risk-sensitive one. A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion (G-SDEs), where the control variable may influence all terms. We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost. Initially, we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework. Subsequently, we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation. Furthermore, we simplified the G-adjoint process from a dual-component structure to a singular component. Moreover, we explained the necessary optimality conditions for this model by considering a convex set of admissible controls. To describe the main findings, we present two examples: the first addresses the linear−quadratic problem and the second examines a Merton-type problem characterized by power utility.

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“…To the best of our knowledge, the derivation of a risk-averse PMP was attempted firstly in [19], where appropriate adjoint equations and maximality conditions formulated in terms of the so-called G-Stochastic calculus are introduced in order to cope with the presence of risk measures. This framework was originally introduced by Peng [14], and developed by the stochastic control community later on, see e.g.…”

Section: Introductionmentioning

confidence: 99%

First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

Bonalli¹,

Bonnet²

2022

Preprint

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In this paper, we derive a set of first-order Pontryagin optimality conditions for a risk-averse stochastic optimal control problem subject to final time inequality constraints, and whose cost is a general finite coherent risk measure. Unlike previous contributions in the literature, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. Moreover, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian.

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A stochastic maximum principle for processes driven by G‐Brownian motion and applications to finance

Cited by 7 publications

References 36 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

<i>G</i>-stochastic maximum principle for risk-sensitive control problem and its applications

First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

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