本篇硕士学位论文研究在金融学中用来描述利率随机演化的\ Vasicek 模型的参数估计问题. 该模型由下面的随机微分方程定义: $dY(t)=(c+bY(t))dt+\sigma dW(t)$, 其中\ $W(t)$ 是布朗运动, 而\ $\sigma> 0,$ $b< 0$ 和\ $c> 0$ 是参数. 基于离散时间样本, 我们给出了线性漂移系数\ $b$ 和常漂移系数\ $c$ 的条件最小二乘估计量\ $\hat{b}_n$ 和\ $\hat{c}_n$. 我们证明\ $\hat{b}_n$ 和\ $\hat{c}_n$ 都是强相合且渐近正态的, 后一性质表明当\ $n\to \infty$ 它们都以\ $n^{-1/2}$ 的速度收敛到对应的参数. 本文还给出了\ $b$ 和\ $c$ 的基于连续时间样本的估计量\ $\tilde{b}_t$ 和\ $\tilde{c}_t$, 它们既是条件最小二乘估计, 又是极大似然估计. 我们还证明了\ $\tilde{b}_t$ 和\ $\tilde{c}_t$ 的强相合性和渐近正态性。

In this Master's degree thesis, we study the Vasicek model, which is used to describe the stochastic evolution of interest rates in finance. The model is defined by the stochastic differential equation: $dY(t)= (c+bY(t))dt + \sigma dW(t)$, where $\sigma> 0,$ $b< 0$ and $c> 0$ are parameters. We give the conditional least squares estimators $\hat{b}_n$ and $\hat{c}_n$ of the linear drift coefficient $b$ and the constant drift coefficient $c$ based on discrete time samples. We show that both $\hat{b}_n$ and $\hat{c}_n$ are strongly consistent and asymptotically normal. The later property means that they both converge to the corresponding parameters at the speed of\ $n^{-1/2}$ as $n\to \infty$. We also give the estimators $\tilde{b}_t$ and $\tilde{c}_t$ of $b$ and $c$ based on continuous time samples, which are conditional least squares estimators as well as maximum likelihood estimators. We also show that $\tilde{b}_t$ and $\tilde{c}_t$ are also strongly consistent and asymptotically normal.