Stochastic integration for set-indexed processes
Let \(\mathcal A\) be a semilattice of closed subsets of a locally compact, complete metric space \(T\), satisfying some additional assumptions, and let \(r({\mathcal A})\) denote the algebra generated by \(\mathcal A\). Given a probability space \((\Omega, {\mathcal F}, P)\), let \(X=\{X(A), A\in r({\mathcal A})\}\) be a real-valued, set-indexed process, such that the function \(\lambda_X(F\times I)=E(1_FX(I))\), \(F\in \mathcal F\), \(I\in r({\mathcal A})\) is a real measure with respect to each variable. For a bounded, \(T\)-indexed process \(Y\) of the form \(Y(\omega,t)=f(t)\Phi(\omega)\) the integral \(\int_TYdX\) is defined as the Radon-Nikodym derivative \(dm/dP\) of the measure \(m(F)=\int_F\int_Tf(t)\Phi(\omega)d\lambda_X(\omega,t)\). It is proved that this definition can be extended to more general \(Y\) by approximation. Next, an analogous integral is defined for set-indexed \(Y\), and finally a localized version of the integral is introduced.
scientific article; zbMATH DE number 1506587