Exponential Functionals of Brownian Motion and Related Processes
Exponential Functionals of Brownian Motion and Related Processes
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DOI：
10.1007/978-3-642-56634-9
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发表时间：
2001-09
期刊：
影响因子：
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通讯作者：
M. Yor
中科院分区：
文献类型：
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作者：
M. Yor
这本专著包含：-十篇论文写的作者，和合作者，在1988年12月和1998年10月之间关于布朗运动和相关过程的某些指数泛函，这已经是，现在仍然是，感兴趣的，至少在过去的十年中，研究人员在数学金融;-介绍了这个问题的观点，数学金融的H。杰曼起源我的兴趣在布朗运动的指数的研究与金融数学的问题，首先问我的S。Jacka于1988年12月在沃里克，后来由M. Chesney在日内瓦和H. Geman在巴黎的一个工作，计算亚式期权的价格，即：尽可能给出下式的显式：（1）其中A~ v）= I~ dsexp ~ 2（Bs+ li S），（Bs，s：0）为实值布朗运动.由于具有漂移的布朗运动的指数过程，通常称为：几何布朗运动，可以表示为：t：0，（2）其中（Rt），u：0）表示一个15维Bessel过程，其中δ = 2（1 I + 1），显然，从（2）[它类似于Feller用布朗运动表示线性扩散X，通过X的标度函数和速度测度，应该可以计算与（1）有关的量，特别是：
This monograph contains:-ten papers written by the author, and co-authors, between December 1988 and October 1998 about certain exponential functionals of Brownian motion and related processes, which have been, and still are, of interest, during at least the last decade, to researchers in Mathematical finance;-an introduction to the subject from the view point of Mathematical Finance by H. Geman. The origin of my interest in the study of exponentials of Brownian motion in relation with mathematical finance is the question, first asked to me by S. Jacka in Warwick in December 1988, and later by M. Chesney in Geneva, and H. Geman in Paris, to compute the price of Asian options, ie: to give, as much as possible, an explicit expression for:(1) where A~ v)= I~ dsexp2 (Bs+ liS), with (Bs, s::::: 0) a real-valued Brownian motion. Since the exponential process of Brownian motion with drift, usually called: geometric Brownian motion, may be represented as: t::::: 0,(2) where (Rt), u::::: 0) denotes a 15-dimensional Bessel process, with 5= 2 (1I+ 1), it seemed clear that, starting from (2)[which is analogous to Feller's repre sentation of a linear diffusion X in terms of Brownian motion, via the scale function and the speed measure of X], it should be possible to compute quan tities related to (1), in particular: in hinging on former computations for Bessel processes.