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2016 | 36 | 2 | 181-206
Tytuł artykułu

A general class of McKean-Vlasov stochastic evolution equations driven by Brownian motion and Lèvy process and controlled by Lèvy measure

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In this paper we consider McKean-Vlasov stochastic evolution equations on Hilbert spaces driven by Brownian motion and L`evy process and controlled by L`evy measures. We prove existence and uniqueness of solutions and regularity properties thereof. We consider weak topology on the space of bounded Le´vy measures on infinite dimensional Hilbert space and prove continuous dependence of solutions with respect to the Le´vy measure. Then considering a certain class of Le´vy measures on infinite as well as finite dimensional Hilbert spaces, as relaxed controls, we prove existence of optimal controls for Bolza problem and some simple mass transport problems
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  • University of Ottawa, Canada
Bibliografia
  • [1] \bibitem {1} N.U. Ahmed, Optimal control of general McKean-Vlasov stochastic evolution equations on Hilbert spaces and necessary conditions of optimality, Discuss. Math. Diff. Incl. Control and Optim. 35 (2015), 165-195. doi: 10.7151/dmdico.1171
  • [2] N.U. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Cotrol and Optim. 46 (2007), 356-378. doi: 10.1137/050645944
  • [3] N.U. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stochastic Process. Appl. 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X
  • [4] N.U. Ahmed and X. Ding, Controlled McKean-Vlasov equations, Commun. Appl. Anal. 5 (2001), 183-206.
  • [5] N.U. Ahmed, C.D. Charalambous, Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and drviven by relaxed controls, SIAM J. Control and Optim. 51 (2013), 3235-3257. doi: 10.1137/120885656
  • [6] N.U. Ahmed, Systems governed by mean-field stochastic evolution equations on Hilbert spaces and their optimal control, Dynamic Systems and Applications (DSA) 25 (2016), 61-88.
  • [7] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, Vol. 246, Longman Scientific and Technical (U.K., Co-published with John-Wiely & Sons, Inc. New York, 1991).
  • [8] N.U. Ahmed, Stochastic evolution equations on Hilbert spaces with partially observed relaxed controls and their necessary conditions of optimality, Discuss. Math. Diff. Incl. Control and Optim. 34 (2014), 105-129. doi: 10.7151/dmdico.1153
  • [9] E. Cinlar, Probability and Stochastics, Graduate Text in Mathematics, Vol. 261 (Springer, 2011).
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  • [12] D.A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys. 31 (1983), 29-85. doi: 10.1007/BF01010922
  • [13] D.A. Dawson and J. Gärtner, Large Deviations, Free Energy Functional and Quasi-Potential for a Mean Field Model of Interacting Diffusions, Mem. Amer. Math. Soc. 398 (Providence, RI, 1989).
  • [14] N. Dunford and J.T Schwartz, Linear Operators, Part 1 (Interscience Publishers, Inc., New York, 1958).
  • [15] N.I. Mahmudov and M.A. McKibben , Abstract second order damped McKean-Vlasov stochastic evolution equations, Stoch. Anal. Appl. 24 (2006), 303-328. doi: 10.1080/07362990500522247
  • [16] H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907
  • [17] M.J. Merkle, On weak convergence of measures on Hilbert spaces, J. Multivariate Anal. 29 (1989), 252-259. doi: 10.1016/0047-259X(89)90026-2
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  • [19] Y. Shen and T.K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal. 86 (2013), 58-73. doi: 10.1016/j.na.2013.02.029
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Bibliografia
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