The (elliptic) stochastic quantization equation for the (massive) $cosh(beta varphi)_2$ model, for the charged parameter in the $L^2$ regime (i.e. $beta^2 < 4 pi$), is studied. We prove the existence, uniqueness and the properties of the invariant measure of the solution to this equation. The proof is obtained through a priori estimates and a lattice approximation of the equation. For implementing this strategy we generalize some properties of Besov space in the continuum to analogous results for Besov spaces on the lattice. As a final result we show as how to use the stochastic quantization equation to verify the Osterwalder-Schrader axioms for the $cosh (beta varphi)_2$ quantum field theory, including the exponential decay of correlation functions.
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