Viscosity solutions to complex Hessian equations

We study viscosity solutions to complex hessian equations. In the local case, we consider $\mathrm{\Omega }$ a bounded domain in ${\mathbb{C}}^n,$ $\beta$ the standard K"{a}hler form in ${\mathcal{C}}^n$ and $1\le m\le n\mathrm{.}$ Under some suitable conditions on $F,g$, we prove that the equation $(dd^c \varphi)^m\wedge\beta^{n-m}=F(x,\varphi)\beta^n,\ \f=g$on$\pO$ admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum are H"{o}lder continuous then so is the solution. In the global case, let $(X,\omega )$ be a compact hermitian homogeneous manifold where $\omega$ is an invariant hermitian metric (not necessarily K"{a}hler). We prove that the equation $(\omega +d{d}^c\phi {)}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$ has a unique viscosity solution under some natural conditions on $F\mathrm{.}$