Running supremum of Brownian motion in dimension 2: exact and asymptotic results

2020/10/15 by Krzysztof Kȩpczyński, Kȩpczyński, Krzysztof
Economics, Econometrics and Finance · Mathematics · #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.2010.07550

arxiv created 2020/10/15 · openalex publication_date 2020/10/15 · arxiv updated 2020/10/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

This paper investigates π_T(a₁,a₂) = ℙ(sup_t∈[0,T] (σ₁B(t)-c₁t)>a₁, sup_t∈[0,T]( σ₂ B(t)-c₂t)>a₂), where \B(t) : t ≥ 0\ is a standard Brownian motion, with T >0, σ₁,σ₂>0, c₁, c₂∈ℝ. We derive explicit formula for the probability π_T(a₁,a₂) and find its asymptotic behavior both in the so called many-source and high-threshold regimes.

Related

• Exact asymptotics of component-wise extrema of two-dimensional Brownian motion
• The trace of spatial Brownian motion is capacity-equivalent to the unit square
• On a first hit distribution of the running maximum of Brownian motion
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• On the boundary of the support of super-Brownian motion: with appendices