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 noise systems, cancer self-remission and smoking.

翻译：本文研究多变量几何布朗运动原点的稳定性。更具体地，在适当的充分条件下，我们构造了一个李雅普诺夫函数，使得多变量几何布朗运动的原点在概率意义下全局渐近稳定。此外，我们证明这些条件可改写为双线性矩阵不等式（BMI）可行性问题。需要强调的是，本文未假设漂移矩阵与噪声扩散矩阵之间的交换性关系，因此多变量几何布朗运动解的所谓马格努斯表示较为复杂。同时，我们通过文献中多个具体模型对该方法进行实例验证，包括随机线性振子、卫星动力学、惯性系统、对角噪声系统、癌症自缓解及吸烟模型。