New approach to optimal control of stochastic Volterra integral equations

Agram, Nacira; Øksendal, Bernt; Yakhlef, Samia

doi:10.48550/arxiv.1709.05463

Cited by 3 publications

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“…These statements are made more precise in the following brief review, where we recall the basic definition and properties of Hida-Malliavin calculus for Lévy processes. The summary is partly based on Agram and Øksendal [2] and Agram et al [3], [4]. General references for this presentation are Aase et al [1], Benth [6], Lindstrøm et al [9], and the books Hida et al [8] and Di Nunno et al [7].…”

Section: A Brief Review Of Hida-malliavin Calculus For Lévy Processesmentioning

confidence: 99%

A Hida–Malliavin white noise calculus approach to optimal control

Agram

Øksendal

2018

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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The classical maximum principle for optimal stochastic control states that if a controlû is optimal, then the corresponding Hamiltonian has a maximum at u =û. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first order derivative was extended to include an extra BSDE for the second order derivatives.In this paper we present an alternative approach based on Hida-Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second order derivatives.The result is illustrated by an example of a constrained linear-quadratic optimal control. MSC(2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.

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Section: A Brief Review Of Hida-malliavin Calculus For Lévy Processesmentioning

confidence: 99%

A Hida–Malliavin white noise calculus approach to optimal control

Agram

Øksendal

2018

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…We also mention [Jaber et al, 2017] for a treatment of Volterra processes with the state and space dependence of affine form. The recent paper [Agram et al, 2018] deals with optimal stopping of stochastic Volterra integral equations. Stochastic Volterra integral equations in a random field setting and driven by a Lévy basis have been considered in [Chong, 2017] and [Pham and Chong, 2018].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations

2022

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We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have nonlinear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

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“…See e.g. Agram et al [9]. Here we are interested in stochastic differential equations (SDEs) where the coefficients of the system depend upon the whole past.…”

Section: Introductionmentioning

confidence: 99%

Mean-field stochastic control with elephant memory in finite and infinite time horizon

Agram

Øksendal

2019

Stochastics

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Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.• For infinite horizon, we derive sufficient and necessary maximum principles.As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.MSC (2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.

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New approach to optimal control of stochastic Volterra integral equations

Cited by 3 publications

References 0 publications

A Hida–Malliavin white noise calculus approach to optimal control

A Hida–Malliavin white noise calculus approach to optimal control

Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations

Mean-field stochastic control with elephant memory in finite and infinite time horizon

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