  We consider random walks in a uniformly elliptic, balanced, i.i.d. random
environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding
problem of stochastic homogenization of non-divergence form difference
operators. We first derive a quantitative law of large numbers for the
invariant measure, which is nearly optimal. A mixing property of the field of
the invariant measure is then achieved. We next obtain rates of convergence for
the homogenization of the Dirichlet problem for non-divergence form operators,
which are generically optimal for $d\geq 3$ and nearly optimal when $d=2$.
Furthermore, we establish the existence, stationarity and uniqueness properties
of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify
the ergodicity of the environmental process for both the continuous-time and
discrete-time random walks, and as a consequence, we get explicit convergence
rates for the quenched central limit theorem of the balanced random walk.
