Stochastic Differential Inclusions

  • Michał Kisielewicz
Part of the Springer Optimization and Its Applications book series (SOIA, volume 157)


In this chapter properties of stochastic differential inclusions are considered. The results of this chapter extend some result presented in the author monograph on the case of stochastic differential inclusions.


  1. 6.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1976)zbMATHGoogle Scholar
  2. 29.
    Ikeda, N., Watanabe, S.: Stochastic Diferential Equations and Diffusion Processes. North Holland Publishing, Amsterdam (1981)zbMATHGoogle Scholar
  3. 31.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (1984)zbMATHGoogle Scholar
  4. 32.
    Jakubowski, A., Memin, J., Pages, G.: Convergence en Loi des suites d’intégrales stochastiquies sur l’espace D 1 de Skorokhod. Probab. Theory Relat. Fields 81, 111–137 (1989)CrossRefGoogle Scholar
  5. 39.
    Kisielewicz, M.: Set-valued stochastic integrals and stochastic inclusions. Disc. Math. 13, 119–126 (1993)MathSciNetzbMATHGoogle Scholar
  6. 40.
    Kisielewicz, M.: Properties of solutions set of stochastic inclusions. J. Appl. Math. Stoch. Anal. 6, 217–236 (1993)MathSciNetCrossRefGoogle Scholar
  7. 42.
    Kisielewicz, M.: Strong and weak solutions to stochastic inclusions. Banach Center Publ. 32, 227–286 (1995)MathSciNetCrossRefGoogle Scholar
  8. 44.
    Kisielewicz, M.: Weak compactness of solution sets to stochastic differential inclusions with convex right hand sides. Topol. Math. Nonlinear Anal. 18, 149–169 (2001)MathSciNetzbMATHGoogle Scholar
  9. 46.
    Kisielewicz, M.: Backward stochastic differential inclusions. Dynam. Syst. Appl. 16, 121–140 (2007)MathSciNetzbMATHGoogle Scholar
  10. 48.
    Kisielewicz, M.: Stochastic Differential Inclusions and Applications. Springer, Berlin (2013)CrossRefGoogle Scholar
  11. 53.
    Kisielewicz, M.: Approximation theorems for set-valued stochastic integrals. Stoch. Anal. Appl. 36(3), 495–520 (2018)MathSciNetCrossRefGoogle Scholar
  12. 57.
    Kisielewicz, M., Michta, M.: Weak solutions of set-valued stochastic differential equations. J. Math. Anal. Appl. 473, 1026–1052 (2019)MathSciNetCrossRefGoogle Scholar
  13. 73.
    Michta, M.: On weak solutions to stochastic differential inclusions driven by semimartingales. Stoch. Anal. Appl. 22(5), 1341–1361 (2004)MathSciNetCrossRefGoogle Scholar
  14. 75.
    Michta, M., Motyl, J.: Compactness of solutions of second order dynamical systems. Dynam. Cont. Discr. Imp. Syst. Ser. A Math. Anal. 14(4), 525–545 (2007)MathSciNetzbMATHGoogle Scholar
  15. 79.
    Motyl, J.: Stochastic functional inclusion driven by semimartingale. Stoch. Anal. Appl. 16(3), 517–532 (1998)MathSciNetCrossRefGoogle Scholar
  16. 80.
    Motyl, J.: Stability problem for stochastic inclusions. Stoch. Anal. Appl. 16, 933–944 (1998)MathSciNetCrossRefGoogle Scholar
  17. 81.
    Motyl, J.: Existence of solutions of set-valued equation. Bull. PAN 46(4), 419–430 (1998)MathSciNetzbMATHGoogle Scholar
  18. 82.
    Motyl, J.: Viable solutions to set-valued stochastic equations. Optimization 48, 157–176 (2000)MathSciNetCrossRefGoogle Scholar
  19. 83.
    Øksendal, B.: Stochastionc Differential Equations. Springer, Berlin (1998)CrossRefGoogle Scholar
  20. 92.
    Stricker, C.: Loi de Semimartingales et Critères de Compacité, Sem. de Probab. XIX Lectures Notes in Mathematics, vol. 1123. Springer, Berlin (1985)Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Michał Kisielewicz
    • 1
  1. 1.Faculty of MathematicsUniversity of Zielona GóraZielona GóraPoland

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