| Summary: | 博士 === 國立中央大學 === 數學研究所 === 97 === Brownian motion is a rigorous mathematical model (Wiener (1923), Levy (1948),
Ciesielski (1961)) with fruitful applications ranging from biology (Brown (1827)),
physics (Einstein (1905), Mazo (2002)), economy and financial engineering
(Bachelier (1900), Black and Scholes (1973)), to stochastic calculus (Ito (1944)),
among others.
Functional of Brownian motion is also useful in stochastic modeling. This is
particularly true for geometric Brownian motion. For instance, it has been applied to
model prices of stock (page 365 in Karlin and Taylor (1975), Black and
Scholes(1973)), rice (Yoshimoto el al. (1996)), labor (page 363 in Karlin and Taylor
(1975)) and others (Shoji (1995)). See Yor (2001) for more details.
Although geometric Brownian motion has a great variety of applications, it can not
cover all the random phenomena. It is then desirable to find a general model with
geometric Brownian motion as a special model. The purpose of this paper is to
investigate the generalizations of geometric Brownian motion
and its variants.
For the processes mentioned above, we will first study their mathematical properties. Next, we will discuss their applications in financial engineering. In practice, the parameters are unknown and have to be inferred from
realizations of processes. We will present estimation and test procedures.
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