Stochastic Integration in Banach Spaces and Applications.

This thesis is concerned with a theory of stochastic integration in Banach spaces and applications in the area of stochastic differential equations in Banach spaces. The concept of cylindrical stochastic process in Banach spaceis defined, then the notion of a cylindrical Wiener process {W(,t)}(,t(GREATERTHEQ)0)in real separable Banach space, B, and based on a probability space ((OMEGA),A,m) is introduced. Each W(,t) is defined to be a linear mapping from B('*), the topological dual of B, into the linear space of real r and om variables on ((OMEGA),A,m). This process reduces to a B-valued Wiener process under certain conditions. A stochastic integral in B(,1) called the weak stochastic integral is defined with integrator {W(,t)}(,t(GREATERTHEQ)0) for a class of stochastic integr and s (PHI)(s,(omega)) which are for each (s,(omega)) bounded linear operators from B into a real separable Banach space B(,1). The weak stochastic integral (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is a cylindrical stochastic process, i.e., for each t(ELEM){0,t(,1)}, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is a weak distribution or a representative of such. Various theorems concerning properties of the weak stochastic integral are proved. Conditions under which the stochastic integral becomes a B(,1)-valued stochastic process are given. The theory of stochastic integration in Banach spaces is then applied to stochastic differential equations in Banach spaces. The following linear stochastic differential equation (or more precisely, stochastic integral equation) in a real separable reflexive Banach B(,1) is considered. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where A is the generator of a strongly continuous semigroup {T(t)}(,t(GREATERTHEQ)0), y(ELEM)D(A('*)) (domain of A('*)) and X(,0) is the initial weak distribution in B(,1). By a solution to this equation is meant a cylindrical stochastic process with certain continuity properties that satisfies the equation. Theorems on existence and uniqueness of the solutions to the above equation are proved. The solution is shown to be the stochastic integral (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)PhDStatisticsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158401/1/8125068.pd