Projections of spherical Brownian motion

Electronic Communications in Probability

Aleksandar Mijatović¹ 

关键词: non-Lipschitz stochastic differential equation;    skew-product decomposition;    pathwise uniqueness;    Wright-Fisher diffusion;   

DOI  :  10.1214/18-ECP162

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

【 授权许可】

CC BY