A simpler proof of the equivalence between the backwards Ito integral and the Mcshane backwards Ito integral

Abstract

The equivalence between the backwards Ito integral and the McShane backwards Ito integral was previously obtained via the stochastic M-integral. This paper provides a simpler proof of this equivalence that avoids the M-integral entirely and is developed within the deterministic framework of the backwards McShane integral. By leveraging Henstock's Lemma and absolute continuity properties, the proposed approach simplifies the theoretical structure, improves conceptual transparency, and offers a foundation that is readily adaptable to other forms of stochastic integration.