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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.
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https://hal.archives-ouvertes.fr/hal-03150317
Contributor : Guy Fayolle Connect in order to contact the contributor
Submitted on : Saturday, November 12, 2022 - 4:45:37 PM
Last modification on : Wednesday, November 16, 2022 - 3:31:27 AM

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  • HAL Id : hal-03150317, version 3
  • ARXIV : 2102.11754

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Guy Fayolle, Sandro Franceschi, Kilian Raschel. On the stationary distribution of reflected Brownian motion in a non-convex wedge. Markov Processes And Related Fields, 2022, Markov Processes ans Related Fields, 28 (5), pp.32. ⟨hal-03150317v3⟩

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