在這本論文中，我們研究兩個獨立的基本Lévy隨機過程之極值過程。我們考慮的Lévy過程有布朗運動、卜松過程，以及此兩種隨機過程的和。 在第二章，我們對兩個獨立的布朗運動，找出其極值過程的期望值、平方期望值，進一步討論其機率性質：獨立增量、平穩增量、到達時間的分布和隨機積分等。統計分析則有似然比值、最大似然估計及漸近的不偏估計。 第三章探討的是兩個獨立的卜松過程，我們找出其極值過程的期望值、平方期望值、邊際分布及條件分布。機率性質的討論包含獨立增量、平穩增量、跳躍時間的分布和間隔時間的分布。統計分析則是同第二章，有似然比值、最大似然估計及近似的不偏估計。 第四章則是將布朗運動與卜松過程結合，我們對兩者之和所形成新的兩獨立過程，寫出其極值過程的期望值、平方期望值的表示式，討論其獨立增量、平穩增量、到達時間的分布等機率性質，統計分析同前兩章。 第五章，我們將對第二章到第四章的結論作整理與解析。 In this dissertation, we study the maximum and minimum of two independent simple L$acute{mbox e}$vy processes. L$acute{mbox e}$vy processes considered are Brownian motion with drift, Poisson process and their sum. Mathematical or probabilistic properties discussed include stationary, martingale, diffusion, stochastic integral, interarrival time, jumping time, first passage time, etc. Statistical properties investigated cover likelihood ratio, maximum likelihood estimator and asymptotically unbiased estimator. This dissertation is structured as follows. In Chapter 2, we deal with the maximum and minimum of two independent Brownian motions with drifts. Chapter 3 contains the works about the maximum and minimum of two independent Poisson processes. Chapter 4 is covered by the results about the maximum and minimum of two independent sums of Brownian motion with drift and Poisson process. In Chapter 5, we provide conclusions and comparisons.
