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In a 2006 article (\cite{A1}), Allouba gave his quadratic covariationdifferentiation theory for It\^o's integral calculus. He defined the derivativeof a semimartingale with respect to a Brownian motion as the time derivative oftheir quadratic covariation and a generalization thereof. He then obtained asystematic differentiation theory containing a fundamental theorem ofstochastic calculus relating this derivative to It\^o's integral, adifferential stochastic chain rule, a differential stochastic mean valuetheorem, and other differentiation rules. Here, we use this differentiationtheory to obtain variants of the Clark-Ocone and Stroock formulas, with andwithout change of measure. We prove our variants of the Clark-Ocone formulaunder $L^{2}$-type conditions; with no Malliavin calculus, without the use ofweak distributional or Radon-Nikodym type derivatives, and without thesignificant machinery of the Hida-Malliavin calculus. Unlike Malliavin orHida-Malliavin calculi, the form of our variant of the Clark-Ocone formulaunder change of measure is as simple as it is under no change of measure, andwithout requiring any further differentiability conditions on the Girsanovtransform integrand beyond Novikov's condition. This is due to the invarianceunder change of measure of the first author's derivative in \cite{A1}. Theformulations and proofs are natural applications of the differentiation theoryin \cite{A1} and standard It\^o integral calculus. Iterating our Clark-Oconeformula, we obtain variants of Stroock's formula. We illustrate theapplicability of these formulas by easily, and without Hida-Malliavin methods,obtaining the representation of the Brownian indicator$F=\mathbb{I}_{[K,\infty)}(W_{T})$, which is not standard Malliavindifferentiable, and by applying them to digital options in finance. We thenidentify the chaos expansion of the Brownian indicator.

Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock's formulas