Continuity of the Ito stochastic integral in Hilbert spaces

Abstract

There exists a satisfactory limit theory for stochastic integrals driven by finite dimensional semimartingales. The paper is an attempt to construct a corresponding theory in the case of processes with values in infinite dimensional Hilbert spaces. Limit theorems are given for scalar-and tensor-product integrals and for Hilbert-Schmidt integrands. As a corollary, a criterion for convergence or the tensor quadratic variation is established.

Keywords:

• Stochastic integral
• Skorohod topology
• functional convergence
• semimartingales
• Hilbert spaces
• Hilbert-Schmidt operators
• tensor quadratic variation
• AMS 1991 Subject Classification: Primary: 60F17; Secondary: 60H05, 60B11, 60G48