| Abstract: |
| Stochastic Differential Equations (SDEs) are intensively employed across various fields, including biology, finance, and engineering. Many prominent models in these disciplines, such as the $n$-dimensional Lotka-Volterra population systems and the Heston stochastic volatility model, possess unique positive solutions. However, simulating multidimensional SDEs with superlinear coefficients presents significant challenges. Standard numerical methods, like the classical Euler-Maruyama (EM) method, are known to diverge in finite time when coefficients grow super-linearly. Furthermore, most existing explicit methods fail to preserve the structural positivity of the exact solution.This talk proposes and analyzes an explicit, cost-effective numerical scheme: the Positivity Preserving Truncated Euler-Maruyama (PPTEM) method. The core of this method lies in combining a truncation technique to manage superlinear growth with a positivity mapping to ensure the numerical solution remains within the positive cone $\mathbb{R}^n_+$. |
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