We study the first-passage problem for a process governed by a stochastic differential equation (SDE) driven simultaneously by both parametric Gaussian and Lévy white noises. We extend the path integral (PI) method to solve the SDE with this combined noise input and the corresponding fractional Fokker-Planck-Kolmogorov equations. Then, the PI solutions are modified to analyze the first-passage problem. Finally, numerical examples based on Monte Carlo simulations verify the extension of the PI method and the modification of the PI solutions. The detailed effects of the system parameters on the first-passage problem are analyzed.

中文翻译：

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参数高斯和莱维白噪声相结合的随机微分方程的第一遍问题的路径积分法

我们研究由参数高斯和Lévy白噪声同时驱动的随机微分方程（SDE）控制的过程的首过问题。我们扩展了路径积分（PI）方法，利用该组合噪声输入和相应的分数Fokker-Planck-Kolmogorov方程来求解SDE。然后，修改PI解决方案以分析首次通过问题。最后，基于蒙特卡洛模拟的数值示例验证了PI方法的扩展和PI解的修改。分析了系统参数对首次通过问题的详细影响。