The one phase stochastic Stefan problem with random disturbance is expressed by using the stochastic variational inequality. The key idea in studying the existence and uniqueness properties of the solution of a stochastic variational inequality is the introduction of the theory of non-linear stochastic differential equations. The filtering equation under noisy observations is derived by applying the martingale representation technique. By using the finite difference approximation method, the derived non-linear filter equation is realized numerically and compared with simulation results.