On the It\^o-Alekseev-Gr\"obner formula for stochastic differential equations

In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an It\^o process. The introduced formula essentially generalizes both the classical Alekseev-Gr\"obner formula from the literature on deterministic differential equations as well as the classical It\^o formula from stochastic analysis. The proposed It\^o-Alekseev-Gr\"obner formula is a powerful tool for deriving strong approximation rates for perturbations and approximations of stochastic ordinary and partial differential equations.