Vol. 13, No. 3, 2020

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Uniqueness of a three-dimensional stochastic differential equation

Carl Mueller and Giang Truong

Vol. 13 (2020), No. 3, 433–444
Abstract

In order to extend the study of the uniqueness property of multidimensional systems of stochastic differential equations, we look at the following three-dimensional system of equations, of which the two-dimensional case has been well studied: $d{X}_{t}={Y}_{t}\phantom{\rule{0.3em}{0ex}}dt$, $d{Y}_{t}={Z}_{t}\phantom{\rule{0.3em}{0ex}}dt$, $d{Z}_{t}=|{X}_{t}{|}^{\alpha }\phantom{\rule{0.3em}{0ex}}d{B}_{t}$. We prove that if $\left({X}_{0},{Y}_{0},{Z}_{0}\right)\ne \left(0,0,0\right)$ and $\frac{3}{4}<\alpha <1$, then the system of equations has a unique solution in the strong sense.

Keywords
white noise, stochastic differential equations, uniqueness
Primary: 60H10
Secondary: 34F05