| 摘要: | 布朗運動（Brownian motion）是一個實用的數學模型 （Wiener (1923), Levy (1948), Ciesielski (1961)），在生物（Brown(1827)）、物理（Eistein (1905), Mazo (2002)）、經濟與財務工程（Bachelier (1900), Black and Scholes (1973)） 隨機微積分（Ito (1944)）及許多領域上廣為研究及應用，成果豐碩， 影響深 遠。 雖然幾何布朗運動有著多元化的應用，但是無法涵蓋所有的隨機現象。因此推廣幾何布朗運動, 可以擴展適用範圍，此為本文之主要的目的。本文研究下列幾何布朗運動所推廣的隨機過程及其變化型式。 我們將研究此隨機過程之數學性質，討論其在財務工程的應用，並提出參數之統計推論。 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. |
