We consider the challenge of sampling from a target probability distribution and calculating expectations when only stochastic (noisy) estimates of the potential function's gradient are available, rather than the true gradient. We analyze the accuracy, measured by Mean Square Error (MSE), of time averages obtained using the Stochastic Gradient UBU (SG-UBU) algorithm, which is based on underdamped Langevin dynamics.  A novel discrete Poisson equation framework is proposed to dissect the sampling error into bias and variance components. Using this framework, we establish bounds on the MSE for SG-UBU, quantifying its dependence on algorithm parameters like step size, iteration count, problem dimension, potential landscape characteristics, and noise level. The analysis confirms the long-term stability of the numerical solutions generated by SG-UBU. Furthermore, we provide an explicit characterization of the numerical bias introduced by the algorithm, showing it decreases linearly with the step size. The dominant term in this bias is directly related to the variance of the stochastic gradient estimate. These findings are extended to the common mini-batch setting, examining the implications for computational cost and efficiency, especially in parallel computing environments. Numerical results confirm the theoretical analysis of the bias behavior.