  This article is the first part of a two-fold study, the objective of which is
the theoretical analysis and numerical investigation of new approximate
corrector problems in the context of stochastic homogenization. We present here
three new alternatives for the approximation of the homogenized matrix for
diffusion problems with highly-oscillatory coefficients. These different
approximations all rely on the use of an embedded corrector problem (that we
previously introduced in [Canc\`es, Ehrlacher, Legoll and Stamm, C. R. Acad.
Sci. Paris, 2015]), where a finite-size domain made of the highly oscillatory
material is embedded in a homogeneous infinite medium whose diffusion
coefficients have to be appropriately determined. The motivation for
considering such embedded corrector problems is made clear in the companion
article [Canc\`es, Ehrlacher, Legoll, Stamm and Xiang, in preparation], where a
very efficient algorithm is presented for the resolution of such problems for
particular heterogeneous materials. In the present article, we prove that the
three different approximations we introduce converge to the homogenized matrix
of the medium when the size of the embedded domain goes to infinity.
