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Title

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

Authors 1

Affiliations

  1. University of Ottawa, Canada

Abstract

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

Keywords

ENG
McKean-Vlasov stochastic differential equation, Hilbert spaces, existence of optimal controls

Bibliography

  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).
  10. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, 1992). doi: 10.1017/CBO9780511666223
  11. G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Math. Soc. Lecture Note Ser. 229 (Cambridge University Press, London, 1996).
  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
  18. I.I. Gihman and A.V. Skorohod, The Theory of Stochastic Processes, I (Springer-Verlag, New York, Heidelberg Berlin, 1974). doi: 10.1007/978-3-642-61943-4
  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
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2016/36/2
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Pages:
181-206
Main language of publication
English
Received
2016-11-30
Published
2016

DOI: 10.7151/dmdico.1186

Exact and natural sciences