In this paper we present a stationary bivariate minification process with Marshall and Olkin exponential distribution. The process is given by \begin{align*} X_n&=K\min(X_{n-1},Y_{n-1}, \eta_{n1}),\\ Y_n&=K\min(X_{n-1}, Y_{n-1}, \eta_{n2}), \end{align*} where $\{(\eta_{n1},\eta_{n2}),n\geq1\}$ is a sequence of independent and identically distributed random vectors, the random vectors $(X_m,Y_m)$ and $(\eta_{n1},\eta{n2})$ are independent for $m<n$ and $\lambda_1>0$, $\lambda_2>0$, $\lambda_{12}>0$, $K>(\lambda_1+\lambda_2+\lambda_{12})/\lambda_{12}$. The innovation distribution, the joint distribution of random vectors $(X_n,Y_n)$ and $(X_{n-j},Y_{n-j})$, $j>0$, the autocovariance and the autocorrelation matrix are obtained. The unknown parameters are estimated and their asymptotic properties are obtained.