  We prove well-posedness results and a priori bounds on the solution of the
Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in
$\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a
class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular
class of $A$ and $n$ and the conditions on the obstacle ensure that the problem
is nontrapping almost surely. These are the first well-posedness results and a
priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$
and for $A$ and $n$ varying independently of $k$.
  These results are obtained by combining recent bounds on the Helmholtz
equation for deterministic $A$ and $n$ and general arguments (i.e. not specific
to the Helmholtz equation) presented in this paper for proving a priori bounds
and well-posedness of variational formulations of linear elliptic stochastic
PDEs. We emphasise that these general results do not rely on either the
Lax-Milgram theorem or Fredholm theory, since neither are applicable to the
stochastic variational formulation of the Helmholtz equation.
