A standard method in deterministic product (or multiplicative) integration for integrating measures (or w.r.t measures) is to exploit Radon-Nikodym property. This technique does not extend to stochastic product integration w.r.t semimartingales. We introduce in this article a multiplicative operator functional (MOF) method to define stochastic product integrals; these integrals (MOF's) take values in certain Fock types spaces ℋTR and ℋHS. First we clearly detail out our method for stochastic product integration w.r.t (finite order) matrix valued Brownian motion. We next extend this to define MOF's \(\hat X\) (product integrals) of Hilbert-Schmidt- (K2-) valued semimartingales X. A central result is the Peano series type representation of \(\hat X\). This result helps us to establish a stochastic Kato-Trotter formula obtaining the classical (deterministic) formula as a special case of our stochastic version. An initial purpose of product integration is to construct solutions of differential equations. We use our stochastic product integrals to construct a solution of a linear stochastic equation similar to Doleans-Dade-Protter equation. The construction itself proves the existence and uniqueness of the solution. Further extensions will appear elsewhere.