Two Brownian particles with rank-based characteristics and skew-elastic collisions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Fernholz, E. Robert¹  Ichiba, Tomoyuki²  Karatzas, Ioannis^1,3 

[1] Intech Investment Management LLC, Princeton, NJ 08542 USA

[2] Univ Calif Santa Barbara, Dept Stat & Appl Probabil, Santa Barbara, CA 93106 USA

[3] Columbia Univ, Dept Math, New York, NY 10027 USA

关键词: Diffusion;    Local time;    Skew Brownian motion;    Time reversal;    Brownian motion reflected on Brownian motion;   

DOI  :  10.1016/j.spa.2013.03.019

来源: Elsevier

【 摘 要 】

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally friction-less interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems. The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang bang type, which we also discuss in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free