2024/05/09 by Richard F. Bass, Krzysztof Burdzy, Bass, Richard F. +1
Economics, Econometrics and Finance · Physics and Astronomy · #60J65 #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2405.06144
openalex publication_date 2024/05/09 · openalex created_date 2024/05/14 · openalex updated_date 2026/07/28
Consider the Skorokhod equation in the closed first quadrant: Xt=x0+ Bt+∫0t\bf v(Xs) dLs, where Bt is standard 2-dimensional Brownian motion, Xt takes values in the quadrant for all t, and Lt is a process that starts at 0, is non-decreasing and continuous, and increases only at those times when Xt is on the boundary of the quadrant. Suppose \bf v equals (-a1,1) on the positive x axis, equals (1,-a2) on the positive y axis, and \bf v(0) points into the closed first quadrant. Let θi=\arctan ai, i=1,2. It is known that there exists a solution to the Skorokhod equation for all t≥ 0 if and only if θ1+θ2<π/2 and moreover the solution is unique if |a1a2|<1. Suppose now that θ1+θ2<π/2, θ2<0, θ1>-θ2>0 and |a1a2|>1. We prove that for a large class of (a1,a2), namely those for which (log|a1|+log|a2|)/(a1+a2)gt;π/2, pathwise uniqueness for the Skorokhod equation fails to hold.