Stochastic product integral w.r.t. infinite dimensional semimartingale:ii–uniform operator topology case

Abstract

While stochastic product integration w.r.t finite rank Hilbert Schmidt operator (K2- ) valued semimartingales has received reasonable attention, there is no work, to date, dealing with infinite dimensional semimartingales. We initiate here a theory of stochastic product integration of K2- predictable processes w.r.t the exponential of a K2- Brownian motion. To understand the estimations involved, we start with a finite rank Brownian motion β while keeping the integrand in K2 and then we continue with infinite dimellsioilal Brownian motion. In order to apply our stochastic product integral to solve Doleans–Dade – Protter type linear stochastic equations, we also need to, and therefore, define the Ito integral in the space HS. We finally give the stochastic product integral construction of the solution of a linear stochastic equation. All this is done in the uniform operator topology case. The more difficult unbounded operator case is the subject matter of the final part of this series