In this paper, we discuss an original covariance boundary-value problem that captures the effects of white noise on the local behavior near stable periodic orbits of the corresponding deterministic dynamical systems. The methodology relies on suitably constructed adjoint variables to project the dynamics onto locally transversal hyperplanes and ensure the existence of a unique solution. For each such hyperplane, the computed covariance matrix describes an approximately Gaussian, stationary distribution that highlights directions of particular sensitivity to noise. In contrast to previous formulations, the boundary-value problem analyzed in this paper makes predictions in the original state-space variables rather than in terms of a reduced set of local coordinates for hyperplanes perpendicular to the local vector field. The formulation is compatible with the general form of an augmented continuation problem for the software package COCO; a non-adaptive version has been implemented as a general-purpose constructor for the periodic-orbit toolbox included in the COCO release. We illustrate the efficacy of the proposed formulation and toolbox by analyzing noise-induced behavior near limit cycles in two examples of nonlinear dynamical systems with autonomous drift terms. The analysis presented in this study can be used to design safe operating regimes for systems working in stochastic environments, as well as to design optimum working conditions for systems utilizing noise, such as energy-harvesting applications.