In this manuscript, we study the stability of the origin for the multivariate geometric Brownian motion. More precisely, under suitable sufficient conditions, we construct a Lyapunov function such that the origin of the multivariate geometric Brownian motion is globally asymptotically stable in probability. Moreover, we show that such conditions can be rewritten as a Bilinear Matrix Inequality (BMI) feasibility problem. We stress that no commutativity relations between the drift matrix and the noise dispersion matrices are assumed and therefore the so-called Magnus representation of the solution of the multivariate geometric Brownian motion is complicated. In addition, we exemplify our method in numerous specific models from the literature such as random linear oscillators, satellite dynamics, inertia systems, diagonal and non-diagonal noise systems, cancer self-remission and smoking.

翻译：本文研究了多元几何布朗运动原点的稳定性。具体而言，在适当的充分条件下，我们构建了一个李雅普诺夫函数，使得多元几何布朗运动的原点在概率意义下是全局渐近稳定的。此外，我们证明了此类条件可以改写为一个双线性矩阵不等式的可行性问题。需要强调的是，我们未假设漂移矩阵与噪声扩散矩阵之间存在任何交换关系，因此多元几何布朗运动解的所谓马格努斯表示形式较为复杂。此外，我们通过文献中的多个具体模型例证了本方法，包括随机线性振荡器、卫星动力学、惯性系统、对角与非对角噪声系统、癌症自愈及吸烟模型等。