On the stationary distribution of reflected Brownian motion in a wedge: differential properties

Abstract

We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections, drift), we study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies between the angles of the model.

A complicated integral expression for this Laplace transform has recently been obtained by two authors of this paper. In the differentially algebraic case, we provide a simple, explicit integral-free expression in terms of a hypergeometric function. It specializes to earlier expressions in several classical cases: the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case (corresponding to the so-called Dieker-Moriarty conditions on the parameters). This paper thus closes, in a sense, the quest of all “simple” cases.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte’s invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform (or its square) as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.