The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

REFERENCES

• Agarwal , R.P. , Dhage , B.C. , and O'Regan , D. 2003 . The upper and lower solution method for differential inclusions via a lattice fixed point theorem . Dynamic Systems and Applications 12 : 1 – 7 .
   Web of Science ®Google Scholar
• Ahmed , N.U. 1994 . Nonlinear stochastic differential inclusions on Banach spaces . Stochastic Analysis and Applications 12 ( 1 ): 1 – 10 .
   Web of Science ®Google Scholar
• Ahmed , N.U. 1994 . Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions. Marcel Dekker Lecture Notes . Optimal Control of Differential Equations 160 : 1 – 19 .
   Google Scholar
• Ahmed , N.U. 2002 . Optimal control of impulsive stochastic evolution inclusions . Discussions in Mathematics DICO 22 ( 2 ): 155 – 184 .
   Google Scholar
• Ahmed , N.U. , and Xiang , X. 2003 . Differential inclusions on Banach spaces and their optimal control . Nonlinear Functional Analysis and Applications 8 ( 3 ): 461 – 488 .
   Google Scholar
• Ahmed , N.U. , and Xiang , X. 2003 . Optimal relaxed controls for differential inclusions on Banach spaces . Stochastic Analysis and Applications 12 ( 1 ): 1 – 10 .
   Web of Science ®Google Scholar
• Aubin , J.P. , and Cellina , A. 1984 . Differential Inclusions . Noordhoff , Leyden .
   Google Scholar
• Aubin , J.P. , and Da Prato , G. 1998 . The viability theorem for stochastic differential inclusions . Stochastic Analysis and Applications 16 : 1 – 15 .
   Web of Science ®Google Scholar
• Dhage , B.C. 2007 . Monotone increasing multi-valued random operators and differential inclusions . Nonlinear Functional Analysis and Applications 12 ( 3 ): 399 – 419 .
   Google Scholar
• Ding , X. , and Wu , R. 1998 . A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales . Stochastic Processes and Applications 78 : 155 – 171 .
   Web of Science ®Google Scholar
• Heikkila , S. 2007 . Fixed point results for multifunctions in ordered topological spaces with applications to inclusion problems and game theory . Dynamic Systems and Applications 16 : 105 – 120 .
   Web of Science ®Google Scholar
• Halidias , N. , and Kloeden , P.E. 2006 . A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient . . doi: 10.1155/JAMSA/2006/73257.
   Google Scholar
• Halidias , N. , and Michta , M. 2008 . The method of upper and lower solutions of stochastic differential equations and applications . Stochastic Analysis and Applications 16 ( 1 ): 16 – 28 .
   Web of Science ®Google Scholar
• Hiai , F. , and Umegaki , H. 1977 . Integrals, conditional expectations, and martingales of multivalued functions . Journal of Multivariate Analysis 7 : 149 – 182 .
   Web of Science ®Google Scholar
• Hu , Sh. , and Papageorgiou , N.S. 1997 . Handbook of Multivalued Analysis . Vol. I . Kluwer , Dordrecht .
   Google Scholar
• Jacod , J. , and Memin , J. 1980 . Weak and strong solutions of stochastic differential equations: existence and stability . In Lecture Notes in Math . Vol. 851 . Springer , 169 – 212 .
   Google Scholar
• Ladde , G.S. , Lakshmikantham , V. , and Pachpatte , B.G. 1982 . The method of upper, lower solutions and Volterra integral equations . Journal of Integral Equations 4 ( 4 ): 353 – 381 .
   Web of Science ®Google Scholar
• Lakshmikantham , V. 1984. Monotone technology for nonlinear problems. Journal of Mathematics and Physical Science 18(1):65–72.
   Google Scholar
• Karatzas , I. , and Shreve , S. 1988 . Brownian Motion and Stochastic Calculus . Springer Verlag , New York .
   Google Scholar
• Kannan , R. , and Lakshmikantham , V. 1983 . Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions . . (1): 291 – 299 .
   Google Scholar
• Kisielewicz , M. 1997 . Set-valued stochastic integrals and stochastic inclusions . Stochastic Analysis and Applications 15 ( 5 ): 783 – 800 .
   Web of Science ®Google Scholar
• Kisielewicz , M. , Michta , M. , and Motyl , J. 2003 . Set-valued approach to stochastic control. Existence and regularity properties . Dynamic Systems and Applications 12 ( 3–4 ): 405 – 432 .
   Web of Science ®Google Scholar
• Kisielewicz , M. , Michta , M. , and Motyl , J. 2003 . Set-valued approach to stochastic control. Viability and semimartingale issues . Dynamic Systems and Applications 12 ( 3–4 ): 433 – 466 .
   Web of Science ®Google Scholar
• Michta , M. 2002 . Stochastic inclusions with multivalued integrators . Stochastic Analysis and Applications 20 ( 4 ): 847 – 862 .
   Web of Science ®Google Scholar
• Michta , M. , and Motyl , J. 2006 . Convex selections of multifunctions and their applications . Optimization 55 ( 1–2 ): 91 – 99 .
   Web of Science ®Google Scholar
• Michta , M. , and Motyl , J. 2007 . Set valued Stratonovich integral and Stratonovich type stochastic inclusion . Dynamic Systems and Applications 16 : 141 – 154 .
   Web of Science ®Google Scholar
• Motyl , J. 1994 . On the solution of stochastic differential inclusion . Journal of Mathematics and Analytical Applications 192 : 117 – 132 .
   Web of Science ®Google Scholar
• Phelps , R.R. 1989 . Convex Functions, Monotone Operators and Differentiability . Springer Verlag , Berlin .
   Google Scholar
• Protter , P. 1990 . Stochastic Integration and Differential Equations: A New Approach . Springer Verlag , New York .
   Google Scholar
• Rockafellar , R.T. , and Wets , R.J.-B. 1998 . Variational Analysis . Springer Verlag , Berlin .
   Google Scholar
• Rong , S. 1985 . On strong solutions, uniqueness, stability and comparison theorems for a stochastic system with Poisson jumps . In Lecture Notes in Contra Inference Science 75 . \ed{Thoma, M.} (ed.), Springer , Berlin , pp. 352 – 381 .
   Google Scholar
• Zeidler , E. 1985 . Nonlinear Functional Analysis and Its Applications I: Fixed Points Theorems . Springer-Verlag , New York .
   Google Scholar