Convergence and truncation error analysis for the formal realization approach

Develop rigorous convergence results and truncation error bounds for the formal realization approach based on causal (Dupire) derivatives for Dupire-smooth processes driven by continuous semimartingales, so that the series-based constructions outlined in the paper are placed on a firm analytical footing and can be used to obtain finite-dimensional state-space realizations.

Background

The paper outlines a realization framework for Dupire-smooth processes using functional Itô calculus and Chen–Fliess-type expansions. In Section 5, the authors explicitly note that the development is primarily formal and that key analytical components—specifically the convergence of the series and control of truncation errors—are not established.

These analyses are necessary to justify the use of the formal Chen–Fliess-style expansions in constructing finite-dimensional state-space realizations, both bilinear and nonlinear, from the proposed Lie/Hankel rank conditions.