In this paper we construct new examples of minimal
Lagrangian submanifolds in the complex hyperbolic space with
large symmetry groups, obtaining three 1-parameter families
with cohomogeneity one. We characterize these submanifolds as
the only minimal Lagrangian submanifolds in
$\mathbb{C}\mathbb{H}^n$ that are foliated by umbilical
hypersurfaces of Lagrangian subspaces $\mathbb{R}\mathbb{H}^n$
of $\mathbb{C}\mathbb{H}^n$. By suitably generalizing this
construction, we obtain new families of minimal Lagrangian
submanifolds in $\mathbb{C}\mathbb{H}^n$ from curves in
$\mathbb{C}\mathbb{H}^1$ and $(n-1)$-dimensional minimal
Lagrangian submanifolds of the complex space forms
$\mathbb{C}\mathbb{P}^{n-1}$, $\mathbb{C}\mathbb{H}^{n-1}$ and
$\mathbb{C}^{n-1}$. We give similar constructions in the
complex projective space $\mathbb{C}\mathbb{P}^n$.