We study the strong existence and uniqueness of solutions within a Weyl chamber for a class of time-dependent particle systems driven by multiplicative noise. This class includes well-known processes in physics and mathematical finance. We propose a method to prove the existence of negative moments for the solutions. This result allows us to analyze two numerical schemes for approximating the solutions. The first scheme is a $\theta$-Euler--Maruyama scheme, which ensures that the approximated solution remains within the Weyl chamber. The second scheme is a truncated $\theta$-Euler--Maruyama scheme, which produces values in $\mathbb{R}^{d}$ instead of the Weyl chamber $\mathbb{W}$, offering improved computational efficiency.

翻译：本文研究一类由乘性噪声驱动的时变粒子系统在 Weyl 腔内的强解存在唯一性。该类系统涵盖了物理学和数理金融中若干著名过程。我们提出了一种证明解具有负矩存在性的方法。该结果使我们能够分析两种逼近解的数值格式。第一种是 $\theta$-Euler--Maruyama 格式，该格式确保逼近解始终位于 Weyl 腔内。第二种是截断 $\theta$-Euler--Maruyama 格式，该格式在 $\mathbb{R}^{d}$ 而非 Weyl 腔 $\mathbb{W}$ 中取值，从而提高了计算效率。