Stochastic product integration and stochastic equations

Abstract

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 ℋ_TR and ℋ_HS. 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.