A stochastic integral of operator-valued functions

TL;DR

This paper introduces a new Hilbert space-valued stochastic integral for operator-valued functions, generalizing classical Ito integrals and extending stochastic calculus to operator-valued measures.

Contribution

It defines and studies a novel stochastic integral that broadens the scope of stochastic calculus to include operator-valued functions and measures.

Findings

The integral generalizes classical Ito integrals.

It encompasses Ito integrals in Fock space.

Provides a framework for operator-valued stochastic calculus.

Abstract

In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space