We test a proposal by Namiki and collaborators to enforce constraints on Langevin equations and Parisi's technique to calculate correlation functions via Langevin equations. The numerical study of the equivalence between the O(3) model and the bound state of the pure ${\mathrm{CP}}^1$ model, which can be demonstrated analytically, is used to measure the performance of these methods. In particular, as expected, we show that (a) the relations $g_{\mathrm{O}\mathrm{}\left(3\right)\mathrm{}}=2\mathrm{}{g}_{{\mathrm{CP}}^1}$ and $E_{\mathrm{O}\mathrm{}\left(3\right)\mathrm{}}=2\mathrm{}{E}_{{\mathrm{CP}}^1}+2\mathrm{}$ for the couplings and energies hold beyond the classical level and (b) the mass gap as a function of the coupling is the same for both models. In addition we calculate the mass gap for the ${\mathrm{CP}}^1$ model minimally coupled to fermions. All results are compared with those obtained using the multihit Metropolis algorithm.

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