  Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a
variety of applications ranging from the theory of stochastic networks to math
finance, and under general stability conditions, it has a unique stationary
distribution. In such applications, to implement a stochastic optimization
algorithm or quantify robustness of a model, it is useful to characterize the
dependence of stationary performance measures on model parameters. In this work
we characterize parametric sensitivities of the stationary distribution of an
RBM in a simple convex polyhedral cone; that is, sensitivities to perturbations
of the parameters that define the RBM --- namely, the covariance matrix, drift
vector and directions of reflection along the boundary of the polyhedral cone.
In order to characterize these sensitivities we study the long time behavior of
the joint process consisting of an RBM along with its so-called derivative
process, which characterizes pathwise derivatives of RBMs on finite time
intervals. We show that the joint process is positive recurrent, has a unique
stationary distribution, and parametric sensitivities of the stationary
distribution of an RBM can be expressed in terms of the stationary distribution
of the joint process. This can be thought of as establishing an interchange of
the differential operator and the limit in time. The analysis of ergodicity of
the joint process is significantly more complicated than that of the RBM due to
its degeneracy and the fact that the derivative process exhibits jumps that are
modulated by the RBM. The proofs of our results rely on path properties of
coupled RBMs and contraction properties related to the geometry of the
polyhedral cone and directions of reflection along the boundary. Our results
are potentially useful for developing efficient numerical algorithms for
computing sensitivities of functionals of stationary RBMs.
