  We first obtain the interior $C^{1,1}$-regularity and solvability for the
degenerate real Monge-Amp\`ere equation in a bounded, $C^3$-smooth and strictly
convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is
only globally $C^{1,1}$, and the $d$-th root of the nonnegative right-hand side
is globally $C^{0,1}$ and convex after adding $K|x|^2$ for some constant $K$.
Then we establish the interior $C^{1,1}$-regularity and solvability for the
degenerate complex Monge-Amp\`ere equation in a bounded, $C^3$-smooth and
strictly pseudoconvex domain in $\mathbb C^d$, under the global
$C^{1,1}$-regularity assumption on the boundary data and the $d$-th root of the
nonnegative right-hand side. Since the derivatives may blow up along
non-tangent directions at the boundary under our regularity assumptions on the
boundary data, we also estimate the derivatives up to second order in both
problems. Our technique is probabilistic by following Krylov's approach.
  The result in the real case extends N. Trudinger and J. Urbas's interior
$C^{1,1}$-regularity result for the homogeneous case [N. Trudinger and J.
Urbas, Bull. Austral. Math. Soc., 30(3): 321~334, 1984.] in the sense of
considering the nonnegative right-hand side. The result in the complex case
generalizes E. Bedford and B. A. Taylor's interior $C^{1,1}$-regularity result
in a ball [E. Bedford and B. A. Taylor, Invent. Math., 37(1): 1~44, 1976.] by
allowing the domain be any bounded, sufficiently smooth and strictly
pseudoconvex one.
