A computationally challenging and open problem is how to efficiently generate equilibrated samples of constrained walks. We present here a general stochastic approach that allows to produce these samples with their correct statistical weight and without rejections. The method is illustrated for a jump process constrained within a cylindrical channel and forced to reach one of its ends. We obtain analytically the exact probability density function of the jumps and offer a direct method for gathering equilibrated samples of a random walk confined in a channel with suitable boundary conditions. Unbiassed walks of arbitrary length can thus be generated with linear computational complexity - even when the channel width is much smaller than the typical bond length of the unconstrained walk. By profiling the metric properties of the generated walks for various bond lengths we characterize the crossover between weak and strong confinement regimes with great detail.