[Paper Review] On the stationary distribution of reflected Brownian motion in a wedge: differential properties

Mireille Bousquet‐Mélou, Andrew Elvey Price+3 more|arXiv (Cornell University)|Jan 5, 2021

Stochastic processes and financial applications45 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions for the Laplace transform of the stationary distribution of reflected Brownian motion in a wedge to be rational, algebraic, differentially finite, or differentially algebraic, based on linear dependencies between the wedge's angles. It provides a closed-form hypergeometric expression in the differentially algebraic case, unifying and simplifying known classical results such as the skew-symmetric, orthogonal reflections, and sum-of-exponentials cases.

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.

Motivation & Objective

• To characterize the algebraic and differential nature of the Laplace transform of the stationary distribution of semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge.
• To identify explicit linear dependencies between the wedge’s angles that determine whether the Laplace transform is rational, algebraic, differentially finite, or differentially algebraic.
• To provide a simple, integral-free expression for the Laplace transform in the differentially algebraic case using hypergeometric functions.
• To unify and generalize known classical results—such as the skew-symmetric, orthogonal reflections, and sum-of-exponential density cases—under a single framework.

Proposed method

• The authors derive a functional equation satisfied by the Laplace transform of the stationary distribution.
• They apply Tutte’s invariant method from enumerative combinatorics to express the Laplace transform (or its square) as a rational function of a canonical hypergeometric invariant.
• They use conformal mapping and boundary value problems to analyze the structure of the Laplace transform on the Riemann surface of the wedge.
• They transform the functional equation into a q-difference equation and apply Galois-theoretic tools to establish differential transcendence of the solution.
• They perform contour deformation and residue calculus in the complex plane to evaluate complicated integrals, leading to a simplified hypergeometric expression.
• They verify consistency across different wedge angle regimes, including β < π/2, β > π/2, and β = π/2, using analytic continuation and symmetry arguments.

Experimental results

Research questions

• RQ1 Under what conditions on the wedge angle β and reflection angles δ, ε is the Laplace transform of the stationary distribution rational, algebraic, or differentially finite?
• RQ2 When is the Laplace transform differentially algebraic, and what is its explicit closed-form expression in this case?
• RQ3 How do known classical cases—such as skew-symmetric reflection, orthogonal reflections, and sum-of-exponential densities—fit into the broader framework of differentially algebraic Laplace transforms?
• RQ4 What role do the parameters α₁ and α₂, defined via δ, ε, θ, and β, play in determining the algebraic nature of the Laplace transform?
• RQ5 Can the complicated integral expression for the Laplace transform be simplified to an explicit hypergeometric form under specific angle conditions?

Key findings

• The Laplace transform is differentially algebraic if and only if α ∈ ℤ + πℤ/β, and in this case, it admits a simple, integral-free expression in terms of a hypergeometric function.
• The expression reduces to known classical results: in the skew-symmetric case (α = 0), the orthogonal reflections case (δ = ε = π/2), and the Dieker-Moriarty sum-of-exponential case, the hypergeometric form recovers previously established formulas.
• The paper provides a complete classification of the Laplace transform’s algebraic nature based on linear dependencies between the angles δ, ε, and β.
• The canonical invariant derived via Tutte’s method is a hypergeometric function that parametrizes the solution space and enables the rational parametrization of the Laplace transform.
• Differential transcendence is established by transforming the functional equation into a q-difference equation and applying Galois theory to the solution space.
• The final expression for the Laplace transform in the differentially algebraic case is given by I = 32√2 π^(3/2) e^{z₁+z₂−|z|} sinω / √|z|, with ω = δ − (arg z)/2, and |z| = √(z₁² + z₂² + 2z₁z₂ cos β).