Bimeasures and measures induced by planar stochastic integrators
The class of all one parameter processes which are stochastic integrators with respect to predictable processes was characterized by Dellacherie [see \textit{C. Dellacherie} and \textit{P.-A. Meyer}, Probabilités et potentiel. Chapitres V à VIII: Théorie des martingales. (1980; Zbl 0464.60001)] and \textit{K. Bichteler} [Ann. Probab. 9, 49-89 (1981; Zbl 0458.60057)]. The two-parameter case is considerably more difficult. For example, a weak martingale need not be a stochastic integrator. The author characterizes two-parameter stochastic processes that are \(L^ 2\) stochastic integrators in terms of measures and bimeasures defined on the product spaces. It is shown that X is an \(L^ 2\) stochastic integrator iff there exists a bimeasure \(\beta\) (d\(\omega\times dz\), \(d\omega\) '\(\times dz')\) on \({\tilde \Omega}\times {\tilde \Omega}\) where \({\tilde \Omega}=(\Omega \times R_{z_ 0})\), such that \[ E\int_{R_{z_ 0}}\phi dX\int_{R_{z_ 0}}\psi dX=\int_{{\tilde \Omega}}\int_{{\tilde \Omega}}\phi_{\zeta}(\omega)\psi_{\zeta}(\omega ')\beta (d\omega \times dz,\quad d\omega '\times dz'). \] This condition is further specialized to processes \(M^ 1_{s,t}\) that are 1-martingales. Then the author considers a special class of \(L^ 2\) integrators that is named the class of ''measure inducing integrators'' because their ''double Doleans function'' can be extended to a finite measure on some \(\sigma\)- field. It is proved that if \(X_ z\) is a ''measure inducing integrator'' and the conditional independence assumption (F-4) is satisfied then \(X^ 2_ z\) admits a Doob-Meyer-Cairoli decomposition.
scientific article; zbMATH DE number 3996761
scientific article