  We prove three results in this paper. First, we prove for a wide class of
functions $\varphi\in C^2(\mathbb{S}^{n-1})$ and $\psi(X, \nu)\in
C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire,
strictly convex, spacelike hypersurface $M_u$ satisfying
$\sigma_k(\kappa[M_u])=\psi(X, \nu)$ and $u(x)\rightarrow
|x|+\varphi\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when
$k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex,
spacelike hypersurface $M_u$ satisfying $\sigma_k(\kappa[M_u])=\psi(x, u(x))$
and $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ as
$|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire,
strictly convex, downward translating solitons $M_u$ with prescribed asymptotic
behavior at infinity for $\sigma_k$ curvature flow equations. Moreover, we
prove that the downward translating solitons $M_u$ have bounded principal
curvatures.
