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  • 1.
    Agram, Nacira
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics. Univ Oslo, Norway.
    Oksendal, Bernt
    Univ Oslo, Norway.
    Mean-field stochastic control with elephant memory in finite and infinite time horizon2019In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 91, no 7, p. 1041-1066Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Agram, Nacira
    et al.
    University of Oslo, Norway;University Mohamed Knider of Biskra, Algeria.
    Øksendal, Bernt
    University of Oslo, Norway.
    Yakhlef, Samia
    University Mohamed Khider of Biskra, Algeria.
    New approach to optimal control of stochastic Volterra integral equations2019In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 91, no 6, p. 873-894Article in journal (Refereed)
    Abstract [en]

    We study optimal control of stochastic Volterra integral equations(SVIE) with jumps by using Hida-Malliavin calculus.

    • We give conditions under which there exist unique solutions ofsuch equations.

    • Then we prove both a sufficient maximum principle (a verificationtheorem) and a necessary maximum principle via Hida-Malliavincalculus.

    • As an application we solve a problem of optimal consumptionfrom a cash flow modelled by an SVIE.

  • 3.
    Albeverio, Sergio
    et al.
    University of Bonn, Institute of Applied Mathematics.
    Hilbert, Astrid
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Kolokoltsov, Vassili
    University of Warwick, Department of Statistics.
    Uniform Asymptotic Bounds for the Heat Kernel and the Trace of a Stochastic Geodesic Flow2012In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 84, no 2-3, p. 315-333Article in journal (Refereed)
    Abstract [en]

    We analyze the asymptotic behaviour of the heat kernel dened by a stochastically perturbedgeodesic ow on the cotangent bundle of a Riemannian manifold for small time and smalldiusion parameter. This extends WKB-type methods to a particular case of a degenerateHamiltonian. We give uniform bounds for the solution of the degenerate Hamiltonian boundaryvalue problem for small time. The results are exploited to derive two sided estimates andmultiplicative asymptotics for the heat kernel semigroup and its trace.

  • 4.
    Basna, Rani
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Hilbert, Astrid
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Kolokoltsov, Vassili
    University of Warwick, UK .
    An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space2017In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 6-7, p. 967-993Article in journal (Refereed)
    Abstract [en]

    We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-NashEquilibrium for the N-player games.

  • 5.
    Berrhazi, Badr-eddine
    et al.
    Ibn Tofail Univ, Morocco.
    El Fatini, Mohamed
    Ibn Tofail Univ, Morocco.
    Hilbert, Astrid
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Mrhardy, Naoual
    Sultan Moulay Slimane Univ, Morocco.
    Pettersson, Roger
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Reflected backward doubly stochastic differential equations with discontinuous barrier2019In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516Article in journal (Refereed)
    Abstract [en]

    In this paper, we investigate reflected backward doubly stochastic differential equations (RBDSDEs) with a lower not necessarily right-continuous obstacle. First, we establish the existence and uniqueness of a solution to RBDSDEs with Lipschitz drivers. In the second part, we present a comparison theorem and we prove the existence of a minimal solution to the RBDSDE with the continuous driver.

  • 6.
    El Fatini, Mohamed
    et al.
    Ibn Tofail Univ, Morocco.
    Laaribi, Aziz
    Sultan Moulay Slimane Univ, Morocco.
    Pettersson, Roger
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Taki, Regragui
    Ibn Tofail Univ, Morocco.
    Levy noise perturbation for an epidemic model with impact of media coverage2019In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 91, no 7, p. 998-1019Article in journal (Refereed)
    Abstract [en]

    This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Levy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.

  • 7.
    Gheryani, Soumaya
    et al.
    Tunis El Manar University, Tunisia.
    Hiroshima, Fumio
    Kyushu University, Japan.
    Lorinczi, József
    Loughborough University, UK.
    Majid, Achref
    Tunis El Manar University, Tunisia.
    Ouerdiane, Habib
    Tunis El Manar University, Tunisia.
    P(φ)-process for the spin-boson model and functional central limit theorem for the associated additive functionals2017In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 6-7, p. 1104-1115Article in journal (Refereed)
    Abstract [en]

    We construct a random process with stationary increments associated to the hamiltonianof the spin boson model consisting of a component describing the spin and a componentgiven by a Schwartz distribution-valued Ornstein-Uhlenbeck process describing the bosoneld. As consequence, We use a functional integral representation of the Hamiltonian toprove a functional central limit theorem for additive functionals and we derive explicitexpressions of the diusion constant.

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