Including uncertainty sources in multi-objective optimization allows more robust design decisions at the cost of transforming the objective into an expectation. The stochastic multi-gradient algorithm (SMGDA)\cite{mercier} extends the Robbins-Monro approach to the multi-objective case, allowing for the minimization of the expected objectives without having to directly calculate them. However, a bias in the algorithm and the inherent noise in stochastic gradients cause the algorithm to converge to only a subset of the whole Pareto front, limiting its use. We reduce the bias of the stochastic multi-gradient calculation using an exponential smoothing technique and promote the exploration of the Pareto front by adding non-vanishing noise tangential to the front. We prove that this algorithm, Transverse Brownian Motion, generates samples in a concentrated set containing the whole Pareto front. Finally, we estimate the set of Pareto optimal design points using only the sequence generated during optimization while also providing bootstrapped confidence intervals using a nearest-neighbor model calibrated with a novel procedure based on the hypervolume metric. Our proposed method allows for the estimation of the whole of the Pareto front using significantly fewer evaluations of the random quantities of interest when compared to a direct sample-based estimation, which is valuable in the context of costly model evaluations. We illustrate the efficacy of our approach with numerical examples in increasing dimension and discuss how to apply the method to more complex problems.