Sensitivity Analysis for the Stationary Distribution of Reflected Brownian Motion in a Convex Polyhedral Cone

Abstract

Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a variety of applications ranging from the theory of stochastic networks to mathematical 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 paper, 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 and has a unique stationary distribution and that 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 because of 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.

Efficient Steady-State Simulation of High-Dimensional Stochastic Networks

“…During the revision of the paper, we learned that Lipshutz and Ramanan (2021, lemma 7.5) produce a similar result that the derivative with respect to the initial condition contracts when the RBM hits all the faces. Although the result in Lipshutz and Ramanan (2021) holds in a more general setting, that is, RBM in a convex polyhedral cone, a quantitative bound on the contraction size is not directly provided in Lipshutz and Ramanan (2021). In contrast, Lemma 5 provides an explicit expression of the contraction whenever RBM hits a face, and this is necessary for high-dimensional analysis.…”

A Monte Carlo Method for Estimating Sensitivities of Reflected Diffusions in Convex Polyhedral Domains

“…There exists κ Λ (α) < ∞ such that if (h, g) is the solution to the ESP {(d i (α), n i , c i ), i ∈ I} for f ∈ D G (R J ), and, for k = 1, 2, (φ k , η k ) is a solution to the derivative problem along h for ψ k ∈ D(R J ), then for all t < ∞, (6.7) sup We close this section with the following continuity property of the derivative map that will be instrumental in the proof of our main result. In addition to its use in this work, it is also used in [16] to show that the joint reflected diffusion and derivative process is Feller continuous. In contrast to the other results in this section, we explicitly state exactly which assumptions are required for the following theorem.…”

“…Establishing convergence of the approximation is quite subtle and relies on a continuity property of a certain map, called the derivative map. This continuity property is of independent interest; for example, it is used in [16] to prove that a reflected Brownian motion (RBM) in a convex polyhedral cone and its derivatives process are jointly Feller continuous.…”