  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.
