Laplace transforms related to excursions of a one-dimensional diffusion
Laplace transforms related to excursions of a one-dimensional diffusion
复制标题
DOI：
10.2307/3318434
复制
发表时间：
1999-04
期刊：
影响因子：
1.5
通讯作者：
J. Pitman;M. Yor
中科院分区：
文献类型：
--
作者：
J. Pitman;M. Yor
众所周知，由一维布朗运动(BM)导出的多个随机时间的拉普拉斯变换具有双曲函数形式的简单表达式。本文给出了这些结果的统一方法，并用S(1971)的游移理论给出了它们在一维扩散中的推广。见Jeanbrac等人。(1997)关于BM的加法泛函分布的Feynman-Kac公式的相关结果的调查，以及关于一维扩散泛函分布的大量公式的参见Borodin和Salminen(1996)。第2节以表格的形式给出了基本的单变量公式，并附有注释和证明。第三节展示了如何将单变量公式与游程理论的独立性结果相结合，得到各种多元拉普拉斯变换。在BM的情况下，这些结果已被Taylor(1975)应用于过程控制和股票市场价格，以及由Pitman和Yor(1986)应用于平面BM绕组的渐近分布。
It is well known that the Laplace transforms of many random times derived from a onedimensional Brownian motion (BM) admit simple expressions in terms of hyperbolic functions. This paper offers a unified approach to these results, and presents their generalizations for a one-dimensional diffusion, using It6's (1971) excursion theory. See Jeanblanc et al. (1997) for a survey of related results involving the Feynman-Kac formula for the distribution of an additive functional of BM, and see Borodin and Salminen (1996) for a vast array of fomulae for the distribution of functionals of a one-dimensional diffusion. Section 2 presents the basic univariate formulae in a table, together with commentary and proofs. Section 3 shows how the univariate formulae can be combined with independence results from excursion theory to obtain various multivariate Laplace transforms. In the case of BM, these results have been applied to process control and stockmarket prices by Taylor (1975), and to the asymptotic distribution of windings of planar BM by Pitman and Yor (1986).