Set-Valued Stochastic Lebesgue Integral And Representation Theorems

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued stochastic process with respect to the time t. Finally we shall prove the presentation theorem of set-valued stochastic integral and discuss further properties that will be useful to study set-valued stochastic differential equations with their applications.

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

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

[3]  V. Kreinovich,et al.  Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables , 2002 .

[4]  L. Rogers Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

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

[6]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[7]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[8]  Shoumei Li,et al.  Decomposition and representation theorem of set-valued amarts , 2007, Int. J. Approx. Reason..

[9]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[10]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[11]  Michał Kisielewicz QUASI-RETRACTIVE REPRESENTATION OF SOLUTION SETS TO STOCHASTIC INCLUSIONS , 1997 .

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

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

[14]  M. Yor DIFFUSIONS, MARKOV PROCESSES AND MARTINGALES: Volume 2: Itô Calculus , 1989 .

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

[16]  Yukio Ogura,et al.  Central limit theorems for generalized set-valued random variables , 2004 .

[17]  Gerald Beer,et al.  Topologies on Closed and Closed Convex Sets , 1993 .

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

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

[20]  Michał Kisielewicz Existence theorem for nonconvex stochastic inclusions , 1994 .

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

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

[23]  R. Stephenson A and V , 1962, The British journal of ophthalmology.

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

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

[26]  A. C. Thompson,et al.  Theory of correspondences : including applications to mathematical economics , 1984 .

[27]  J. Motyl Existence of solutions of set-valued Ito equation , 1998 .

[28]  M. Kisielewicz Properties of solution set of stochastic inclusions , 1993 .

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

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

[31]  J. Motyl Stability problem for stochastic inclusion , 1998 .

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

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

[34]  Dan A. Ralescu,et al.  Strong Law of Large Numbers for Banach Space Valued Random Sets , 1983 .

[35]  Michał Kisielewicz Viability theorems for stochastic inclusions , 1995 .

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

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

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

[39]  Reg Kulperger,et al.  Minimax pricing and Choquet pricing , 2006 .