  This article introduces the groundbreaking concept of the financial
differential machine learning algorithm through a rigorous mathematical
framework. Diverging from existing literature on financial machine learning,
the work highlights the profound implications of theoretical assumptions within
financial models on the construction of machine learning algorithms.
  This endeavour is particularly timely as the finance landscape witnesses a
surge in interest towards data-driven models for the valuation and hedging of
derivative products. Notably, the predictive capabilities of neural networks
have garnered substantial attention in both academic research and practical
financial applications.
  The approach offers a unified theoretical foundation that facilitates
comprehensive comparisons, both at a theoretical level and in experimental
outcomes. Importantly, this theoretical grounding lends substantial weight to
the experimental results, affirming the differential machine learning method's
optimality within the prevailing context.
  By anchoring the insights in rigorous mathematics, the article bridges the
gap between abstract financial concepts and practical algorithmic
implementations.
