Operator-valued stochastic differential equations in the context of Kurzweil-like equations

The theory of generalized ordinary differential equations (ODEs) is known to be a very powerful theory that not only includes ordinary and functional differential equations, but also includes impulsive and measure differential equations, integral equations as well as dynamical equations on time scales as special cases. The purpose of this paper is to provide a similar environment for operator-valued stochastic differential equations (SDEs). In our approach, we use a modified version of Kurzweil's original definition of his integral by replacing full divisions by belated partial divisions. This allows us to extend the Ito-Henstock integral to include a larger class of SDEs, which can contain highly oscillatory (operator-valued) functions of unbounded variation. We provide a detailed discussion of a few examples which highlight our main results.& COPY; 2023 Elsevier Inc. All rights reserved.