Set-Valued Square Integrable Martingales and Stochastic Integral

In this paper, we firstly introduce the concept of set-valued square integrable martingales. Secondly, we give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale, and then prove the representation theorem of this kind of integral processes. Finally, we show that the stochastic integral process is a set-valued sub-martingale.

[1]  Christian Hess,et al.  On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence , 1991 .

[2]  H. Frankowska,et al.  A stochastic filippov theorem , 1994 .

[3]  Nikolaos S. Papageorgiou On the conditional expectation and convergence properties of random sets , 1995 .

[4]  Yukio Ogura,et al.  A convergence theorem of fuzzy-valued martingales in the extended Hausdorff metric H[infin] , 2003, Fuzzy Sets Syst..

[5]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[6]  Michał Kisielewicz Set-valued stochastic intergrals and stochastic inclutions 1 , 1997 .

[7]  J. Kim,et al.  On Set-Valued Stochastic Integrals , 2003 .

[8]  S. Shreve Stochastic calculus for finance , 2004 .

[9]  Shoumei Li,et al.  ON THE SOLUTIONS OF SET-VALUED STOCHASTIC DIFFERENTIAL EQUATIONS IN M-TYPE 2 BANACH SPACES , 2009 .

[10]  Yukio Ogura,et al.  Convergence of set-valued and fuzzy-valued martingales , 1999, Fuzzy Sets Syst..

[11]  R. Aumann INTEGRALS OF SET-VALUED FUNCTIONS , 1965 .

[12]  Dinh Quang Luu Applications of set-valued Radon-Nikodym theorems to convergence of multivalued $L^1$-amarts. , 1984 .

[13]  J. Kim,et al.  Stochastic Integrals of Set-Valued Processes and Fuzzy Processes , 1999 .

[14]  B. Øksendal Stochastic Differential Equations , 1985 .

[15]  Nikolaos S. Papageorgiou On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation , 1985 .

[16]  Yukio Ogura,et al.  Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense , 1998 .

[17]  Nikolaos S. Papageorgiou On the theory of Banach space valued multifunctions. 2. Set valued martingales and set valued measures , 1985 .

[18]  F. Hiai,et al.  Integrals, conditional expectations, and martingales of multivalued functions , 1977 .

[19]  N. Ahmed Nonlinear stochastic differential inclusions on balance space , 1994 .

[20]  Shoumei Li,et al.  Stochastic integral with respect to set-valued square integrable martingales , 2010 .

[21]  Aihong Ren,et al.  Representation theorems, set-valued and fuzzy set-valued Ito integral , 2007, Fuzzy Sets Syst..

[22]  Shoumei Li,et al.  Strong solution of Itô type set-valued stochastic differential equation , 2010 .

[23]  Shoumei Li,et al.  On set-valued stochastic integrals in an M-type 2 Banach space , 2009 .

[24]  Shouchuan Hu,et al.  Handbook of multivalued analysis , 1997 .

[25]  Sitadri Bagchi On a. s. convergence of classes of multivalued asymptotic martingales , 1985 .

[26]  Zhen Wang,et al.  On convergence and closedness of multivalued martingales , 1994 .

[27]  Ioannis Karatzas,et al.  Lectures on the Mathematics of Finance , 1996 .

[28]  Jean-Pierre Aubin,et al.  The viability theorem for stochastic differential inclusions 2 , 1998 .

[29]  A. Korvin,et al.  A convergence theorem for convex set valued supermartingales , 1985 .

[30]  Shouchuan Hu,et al.  Handbook of Multivalued Analysis: Volume I: Theory , 1997 .

[31]  LI Shi-kai Square Integrable Martingale , 2008 .

[32]  Jungang Li,et al.  Set-Valued Stochastic Lebesgue Integral And Representation Theorems , 2008, Int. J. Comput. Intell. Syst..

[33]  I. Molchanov Theory of Random Sets , 2005 .

[34]  M. Kisielewicz,et al.  Weak Compactness of Solution Sets to Stochastic Differential Inclusions with Non-Convex Right-Hand Sides , 2005 .