On the stationary distribution of reflected Brownian motion in a non-convex wedge

Abstract

We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

“…The present work is a companion paper of [11], where we proved that the three-quarter plane stationary distribution could be found by solving a two-dimensional vector boundary value problem (BVP) for the associated Laplace transforms. Here we go much further, by showing that one can actually reduce the latter to a classical scalar BVP.…”

“…This is however not sufficient, as we need to continue meromorphically the Laplace transform onto an open domain of C 2 containing (0, 0). For that purpose, we introduce a continuation procedure related to the one used in [11], and close to those in [10,13] in the discrete setting. This is done in Section 4 and in the Appendix; the main idea is to split the three-quarter plane into two convex cones, and to state two intermediate functional equations for Laplace transforms whose convergence is clear.…”

“…A related problem was solved in [1], for a diffusion process in the positive quarter plane. For more details, we refer to the introductions of [11,14]. A precise semimartingale definition of reflected Brownian motion will be given in Section 2.…”

“…While the early articles [25,26] most dealt with the general case ζ ∈ (0, 2π ), the case of convex cones ζ ∈ (0, π] has attracted much more attention [1,5,9,[14][15][16]. However, as explained in the introduction of [11], there are numerous reasons to look at the non-convex, Fig. 2.1.…”

“…• However, when the cone is not convex, some integral transforms will not converge, and one has to divide the three-quarter plane into two convergence regions. This leads to a system of two functional equations for the Laplace transforms, see [11], which is more awkward to solve.…”