A first-passage-place problem for integrated diffusion processes

Pub Date : 2023-05-22 DOI:10.1017/jpr.2023.19
M. Lefebvre
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Abstract

Let ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$ , where Y(t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived.
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积分扩散过程的一个首次通过位置问题
设${\mathrm{d}}X(t)=-Y(t)\,{\mathrm{d}t$,其中Y(t)是一维扩散过程,设$\tau(X,Y)$是从(X,Y)开始的过程第一次离开第一象限的子集。概率$p(x,y)\,:\!=\的计算问题,\mathb{P}[X](\tau(X,y))=0]$。在重要的特殊情况下,得到了函数p(x,y)的拉普拉斯变换,并表明该变换至少可以在数值上反演。也可以导出$\mathbb{E}[\tau(x,y)]$的拉普拉斯变换和$\tau(x,y)$的矩生成函数的显式表达式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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