Diffusions with rank-based characteristics and values in the nonnegative quadrant

TL;DR

This paper constructs and analyzes rank-based diffusions in the nonnegative quadrant with reflection, local drift, and variance characteristics, addressing existence, uniqueness, and boundary behavior.

Contribution

It introduces a novel construction of rank-based diffusions with reflection in the nonnegative orthant, including cases where variances may vanish, and studies their pathwise properties.

Findings

Corner of the quadrant is never visited if laggard's variance ≥ leader's variance.

Constructs diffusions via coupled Skorokhod reflection equations.

Addresses pathwise uniqueness and strength of the associated stochastic differential equations.

Abstract

We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then ``unfolding'' the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534-536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited.