Stochastic Quadratic Covariation Differentiation Theory

For a little more than sixty years the classical Ito calculus existed only as an integral calculus without a definition of a stochastic derivative of a semimartingale with respect to another semimartingale that's an anti Ito-integral and that leads to a corresponding systematic theory of pathwise differentiation.  In 2004 I completed the elements of this theory (including a fundamental theorem of stochastic calculus, a differential chain rule, a differential mean-value theorem, other differentiation rules and formulas, and more), and parts of this work are published in my "A Differentiation Theory for Ito's Calculus" 2006 paper. 

1. The linked journal reference is:  A Differentiation Theory for Ito's Calculus     Stochastic Analysis and Applications , Vol. 24, No. 2, (2006),   Abstract     
   

2. The linked arXiv version of my article is:   A Differentiation Theory for Itô's Calculus