SEQUENTIAL ANALYSIS FOR FINANCIAL TIME SERIES

我们在这个项目中考虑了两个问题。第一个是由正态随机游走中的变点问题引起的技术问题。设x_1，x_2，…我们设S_n=x_1是一个独立的正态分布的均值为1的正态分布随机变量序列。X_n为n*1。假设H_0：Theta=0相对于备选H_1：Theta&gt；0的序贯检验包括拒绝H_0而支持H_1当且仅当t*m，其中停止时间t=inf{n*1；S_n*(2a(n，c))^&lt；1/2&gt；}定义为常数a&gt；0，c*0，且m为正整数。利用非线性更新理论，Siegmund(1977,1978，Biilluska)计算出t(M)=min{t，m}在不同的theta值下的期望值极限为m=m(A)**，a**，使得2a/m是固定的(=theta_0)。我们计算了所有theta=theta_0(1u/(2a)^&lt；1/2&gt；)的E_*{t(M)}的渐近展开式。我们让a通过theta_0^2/2的整数倍变为无穷大，使得m=2a/theta_0^2是整数。对于每个theta，我们设N=[2a/theta^2]和Rho=(2a/theta^2)-N。我们的主要结果在Takahashi(1993)的定理2中给出，其中常量Rho起着重要的作用。我们从随机选取的100只股票的时间序列数据中提取了几个因素，然后对这些因素进行AR模型拟合。最后采用卡尔曼滤波对模型中的参数进行估计。其结果见于Takubo，Tanaka和Takahashi(1993)。

We considered two problems in this project. The first one is a technical one arising from the change point problem in the normal random walk. Let x_1, x_2,... be a sequence of independent and normally distributed random variables with mean theta and variance 1. We let s_n = x_1+...+x_n for n * 1. A sequential test for the hypothesis H_0:theta =0 against the alternative H_1 : theta >0 consists of rejecting H_0 in favor of H_1 if and only if t * m, where the stopping time t = inf{n* 1; s_n* (2a(n+c))^<1/2>} is defined for constants a>0, c * 0, and m is a positive integer. Using the non-linear renewal theory Siegmund(1977,1978,Biometrika) calculate the limits of the expected value of t(m) = min{t,m} at the various theta values as m=m(a) ** , a ** in such a way that 2a/m is fixed (=theta_0). We calculated asymptotic expantions for E_*{t(m)} for all theta =theta_0(1+u/(2a)^<1/2>). We let a goes to infinity throught the integral multiple of theta_0^2/2, so that m=2a/theta_0^2 are integer. For each theta , we let N = [2a/theta^2] and rho = (2a/theta^2) - N. Our main results is given in theorem 2 of Takahashi(1993), where the constant rho playes an important role.Another problem we considered in this project is to analyze daily Nikkei 225 for 3 years from 1987 to 1989. We extract several factors from the randomly selected 100 stock's time series data, and then fit AR model to these factors. Finally we adopt Kalman filter to estimate the parameters in the model. The results are found in Takubo, Tanaka and Takahashi(1993).