Some Results on Backward Stochastic Differential Equations of Fractional Order

• Nazim I. Mahmudov ^[1] ; Arzu Ahmadova ^[2]
  1. [1] Eastern Mediterranean University

     Eastern Mediterranean University

     Chipre

     
  2. [2] Eastern Mediterranean University & University of Duisburg-Essen
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
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• Resumen

  • In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order α ∈ ( 1 2 , 1) whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.

• Referencias bibliográficas
  • 1. Baghani, O.: On fractional Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simulat. 42, 675–681 (2017)....
  • 2. Bensoussan, A.: Lectures on stochastic control. In: Mittler, S.K., Moro, A. (eds.) Nonlinear filtering and stochastic control, pp. 1–62....
  • 3. Bismut, J.M.: Theorie probabiliste du controle des diffusions. Mem. Amer. Math. Soc. 176, 1–30 (1973)
  • 4. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge (1992)
  • 5. Dauer, J.P., Mahmudov, N.I., Matar, M.M.: Approximate controllability of backward stochastic evolution equations in Hilbert spaces. J....
  • 6. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler functions. Springer-Verlag, Berlin, Related Topics and Applications...
  • 7. Hu, Y., Peng, S.: Adapted solutions of a backward semilinear stochastic evolution equation. Stochastic Anal. Appl. 9(4), 445–459 (1991)
  • 8. Itô, K.: Stochastic differential equations. Mem. Amer. Math. Soc. 4, 1–51 (1951)
  • 9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier Sceince B.V. 1,...
  • 10. Lin, J.: Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stochastic Anal. Appl. 20(1), 165–183 (2002)
  • 11. Mahmudov, N.I., McKibben, M.A.: On backward stochastic evolution equations in Hilbert spaces and optimal control, Nonlinear. Analysis...
  • 12. Mao, X.: Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stoch. Process. Appl. 58, 281–292...
  • 13. Oksendal, B.: Stochastic differential equations: an introduction with applications. Springer-Verlag, Heidelberg (2000)
  • 14. Oldham, K.B., Spanier, J.: The fractional calculus. Academic Press, San Diego (1974)
  • 15. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61 (1990)
  • 16. Pardoux, E., Rascanu, A.: Backward stochastic variational inequalities. Stoch. Stoch. Rep. 67(3–4), 159–167 (1999)
  • 17. Peng, S.: Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27, 125–144 (1993)
  • 18. Peng, S.: A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics Stochastics Rep. 38, 119–134 (1992)
  • 19. Peng, S.: Probabilistic interpletation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 32, 61–74...
  • 20. Rong, S.: On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209–236 (1997)
  • 21. Rong, S.: On solutions of backward stochastic differential equations with jumps and with nonLipschitzian coefficients in Hilbert spaces...
  • 22. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)
  • 23. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York...
  • 24. Shi, Y., Wang, T.: Solvability of general backward stochastic Volterra integral Equations. J. Korean
  • Math. Soc. 49(6), 1301–1321 (2012) 25. Shi, Y., Wen, J., Xiong, J.: Backward doubly stochastic Volterra integral equations and their applications....
  • 26. Tang, S., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control. Optim. 32, 1447–1475...
  • 27. Tessitore, G.: Existence, uniqueness, and space regularity of the adapted solutions of a backward SPDE. Stochastic Anal. Appl. 14(4),...
  • 28. Wang, T., Yong, J.: Backward stochastic Volterra integral equations-Representation of adapted solutions. Stoch. Process. Appl. 129, 4926–4964...
  • 29. Yong, J.: Well-posedness and regularity of backward stochastic Volterra integral equations. Probab. Theory Related Fields 142, 21–77 (2008)
  • 30. Yong, J.: Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779–795 (2006)