On a class of backward stochastic Volterra integral equations

Djordjević, Jasmina; Janković, Svetlana

doi:10.1016/j.aml.2013.07.006

Cited by 28 publications

(15 citation statements)

References 12 publications

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“…(where Y (•) could be higher dimensional) was firstly studied by Lin [23], followed by several other researchers: Aman and N'Zi [2], Wang and Zhang [41], Djordjević and Janković [9,10], Hu and Øksendal [17], etc. Inspired by the study of optimal control problems for forward stochastic Volterra integral equations (FSVIEs, for short), Yong [44] introduced more general BSVIEs, together with the notion of adapted M-solution.…”

Section: Bsviesmentioning

confidence: 99%

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Wang

Yong

2021

ESAIM: COCV

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An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal values $Z(r,r)$ of $Z(\cd\,,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly discussed.

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Section: Bsviesmentioning

confidence: 99%

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Wang

Yong

2021

ESAIM: COCV

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“…This type of generalization of diffusion coefficient has been proved by Janković, Djordjević, Jovanović for backward doubly stochastic differential equations (shorter BDSDEs) under different conditions for the coefficients of the equation in [3] and [5]. Also, in [4] Djordjević and Janković gave the result of this type of extension for the backward stochastic Volterra integral equations (shorter BSVIEs).…”

Section: Extension Of the Class Of Delayed Backward Stochastic Differ...mentioning

confidence: 72%

“…Proof. Our method follows the same idea used by Janković, Ðor dević and Jovanović [3,4,5]. In this fact, let us consider this following BSDE…”

Section: Extension Of the Class Of Delayed Backward Stochastic Differ...mentioning

confidence: 99%

General Fully Coupled Forward Backward Stochastic Differential Equations with delayed generator

Aman¹,

Coulibaly²,

Djordjevic³

2021

Preprint

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This paper is devoted to study different type of BSDE with delayed generator. We first establish an existence and uniqueness result under delayed Lipschitz condition for non homogenous backward stochastic differential equation with delayed generator. Next, existence and uniqueness result for general fully coupled forward backward stochastic differential equation with delayed coefficients has been derived.

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“…Since their introduction, BSVIEs have been extended to much more general frameworks than the one presented above. Hence, Wang [57] studies the case of random Lipschitz data; Wang and Zhang [66] and Shi and Wang [47] deal with general non-Lipschitz data; Coulibaly and Aman [19] study time-delayed generators; mean-field BSVIEs are considered in Shi, Wang, and Yong [48]; Lu [38], Hu and Øksendal [30] and Popier [42] studied extensions to general filtrations and the case where B is replaced by more general processes; Djordjević and Janković [22,21] were interested in perturbed BSVIEs, i.e. when the coefficients depend additively on small perturbations; BSVIEs in Hilbert spaces have been investigated in Anh and Yong [5], Anh, Grecksch, and Yong [6], and Ren [45]; and an analysis of numerical schemes for BSVIEs has been proposed in Bender and Pokalyuk [8].…”

Section: Introductionmentioning

confidence: 99%

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

Hernández

Possamaï

2021

Electron. J. Probab.

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We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semi-linear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of timeinconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

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On a class of backward stochastic Volterra integral equations

Cited by 28 publications

References 12 publications

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

General Fully Coupled Forward Backward Stochastic Differential Equations with delayed generator

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

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