On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts
On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts
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DOI：
10.1239/aap/1046366105
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发表时间：
2003-03
影响因子：
1.2
通讯作者：
H. Matsumoto;M. Yor
中科院分区：
文献类型：
--
作者：
H. Matsumoto;M. Yor
用α t（μ）表示布朗运动{B，s ≥ 0}的At（μ）= λ 0 texp（2（B +μ s））ds的概率律.众所周知，α t（μ）在金融数学、随机环境中的扩散过程、双曲空间上的随机分析等许多领域都有重要意义，但它的表达式比较复杂。最近，Dufresne得到了α t（0）和α t（1）的一些非常简单的表达式，以及α t（μ）和α t（ν）之间对于两种不同漂移μ和ν的同样显著的关系。本文结合前人关于α t（μ）的一些结果，给出了Dufresne结果的不同证明，并推广到了过程{At（μ），t ≥ 0}.
Denote by α t (μ) the probability law of A t (μ) = ∫0 t exp(2(B s +μ s))ds for a Brownian motion {B s , s ≥ 0}. It is well known that α t (μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for α t (0) and α t (1), as well as an equally remarkable relationship between α t (μ) and α t (ν) for two different drifts μ and ν. In this paper, hinging on previous results about α t (μ), we give different proofs of Dufresne's results and present extensions of them for the processes {A t (μ), t ≥ 0}.