The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions

Chighoub, Farid; Mezerdi, Brahim

doi:10.1155/2014/201491

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…Since the relationship between these two approaches is the one between the derivatives of the value function and the adjoint processes along the optimal state, or actually the one between HJB equations and stochastic Hamiltonian systems, and more generally, the one between PDEs and SDEs. For recent development of the relationship between dynamic programming and maximum principle for stochastic optimal control problems (without delay but including jump diffusions, Markov switching, singular control, or FBSDE systems), refer to Framstad et al [15], Shi and Wu [34], Donnelly [9], Zhang et al [42], Bahlali et al [4], Shi and Yu [35], Chighoub and Mezerdi [8]. Thereby, it is natural to ask the question: Are there any relations between these two extensively used and important approaches, for stochastic optimal control problems with time delay?…”

mentioning

confidence: 99%

Stochastic recursive optimal control problem with time delay and applications

Shi¹,

Jin²,

Zhang³

2015

Mathematical Control &Amp; Related Fields

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This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.

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confidence: 99%

Stochastic recursive optimal control problem with time delay and applications

Shi¹,

Jin²,

Zhang³

2015

Mathematical Control &Amp; Related Fields

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“…As mentioned above, two problems must be considered. The first is optimizing this system by invoking Bellman dynamic programming theorem, Pontryagain's stochastic optimization theorem, and Fleming theorem, then transferring the optimal problem to a corresponding Hamilton-Jacobi-Isaacs equation, which has been resolved respectively by Chighoub et al [10], Guo et al [11], Gomoyunov [12]. The second one is constructing a payoff distribution procedure for all agents [13,14].…”

Section: Behavior and The Equilibrium Of The Agent In Multi-local-wor...mentioning

confidence: 99%

Optimal Strategies Trajectory with Multi-Local-Worlds Graph

Wang¹,

Zheng²,

Zeng³

et al. 2023

Computers, Materials &Amp; Continua

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This paper constructs a non-cooperative/cooperative stochastic differential game model to prove that the optimal strategies trajectory of agents in a system with a topological configuration of a Multi-Local-World graph would converge into a certain attractor if the system's configuration is fixed. Due to the economics and management property, almost all systems are divided into several independent Local-Worlds, and the interaction between agents in the system is more complex. The interaction between agents in the same Local-World is defined as a stochastic differential cooperative game; conversely, the interaction between agents in different Local-Worlds is defined as a stochastic differential non-cooperative game. We construct a non-cooperative/cooperative stochastic differential game model to describe the interaction between agents. The solutions of the cooperative and noncooperative games are obtained by invoking corresponding theories, and then a nonlinear operator is constructed to couple these two solutions together. At last, the optimal strategies trajectory of agents in the system is proven to converge into a certain attractor, which means that strategies trajectory are certainty as time tends to infinity or a large positive integer. It is concluded that the optimal strategy trajectory with a nonlinear operator of cooperative/noncooperative stochastic differential game between agents can make agents in a certain Local-World coordinate and make the Local-World payment maximize, and can make the all Local-Worlds equilibrated; furthermore, the optimal strategy of the coupled game can converge into a particular attractor that decides the optimal property.

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“…See also the recent paper by ∅ksendal and Sulem [29], where Malliavin calculus techniques have been used to define the adjoint process. The relationship between the stochastic maximum principle and dynamic programming has been investigated in [5,15]. See also [28] for some worked examples.…”

Section: Introductionmentioning

confidence: 99%

A stochastic maximum principle for mixed regular-singular control problems via Malliavin calculus

Mezerdi

Yakhlef

2015

Afr. Mat.

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Our main concern in this paper is to study mixed regular-singular control problems, where the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state process, as well as the running and final costs are random functions, so as the state process is no longer a Markov process. Our main result is to derive necessary conditions for optimality, also known as the Pontriagin stochastic maximum principle by using Malliavin calculus techniques. The adjoint process, which plays a key role in the stochastic maximum principle, is given by means of the Malliavin derivatives of the optimal state process.

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The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions

Cited by 5 publications

References 29 publications

Stochastic recursive optimal control problem with time delay and applications

Stochastic recursive optimal control problem with time delay and applications

Optimal Strategies Trajectory with Multi-Local-Worlds Graph

A stochastic maximum principle for mixed regular-singular control problems via Malliavin calculus

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