1. wiener Processes （维纳过程）
Stochastic processes(随机过程） ： Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. Stochastic processes can be classified as discrete time or continuous time.
随机过程就是变量根据时间变化，而这个变化是不可预测的。随机过程分为离散时间和连续时间两种。
Markov Process(马尔科夫过程） ： a particular type of stochastic process whre only the current value of a variable is relevent for predicting the future.
马尔科夫过程是特殊的随机过程，只有当前变量值会影响后面的值，历史数据对后面的预测没有影响。拿到股票上面就是只有当前的股票价格对后续价格有影响，当前价格已经包含了历史数据的信息。
这符合weak form of market efficiency（弱式市场假说），如果历史数据可以预测后面的股价，那么很多人会据此获利，去购买某些预测会涨的股票，这些股票价格会飙升，破坏原有的pattern，达到一个新的平衡
Wiener Processes（维纳过程） ： It’s a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year.
维纳过程就是我们上述的一年变动符合N（0,1）的特殊马尔科夫过程
- 公式如下，Δz 是变动，ε符合标准正态分布N（0,1），√Δt是时间变动
于是Δz是符合马尔科夫过程的
import numpy as np import matplotlib.pyplot as plt import math daysCount = 1000 days = np.arange(0, daysCount, 1) changes = np.random.randn(daysCount - 1, 1)*np.sqrt(1/daysCount) prices = np.zeros(days.shape) prices[0] = 10 for index in range(changes.shape[0]): prices[index + 1] = prices[index] + changes[index][0] plt.plot(days, prices) plt.show() 
import numpy as np import matplotlib.pyplot as plt import math #Wiener Process daysCount = 1000 days = np.arange(0, daysCount, 1) changes = np.random.randn(daysCount - 1, 1)*np.sqrt(1/daysCount) prices = np.zeros(days.shape) prices[0] = 10 for index in range(changes.shape[0]): prices[index + 1] = prices[index] + changes[index][0] plt.plot(days, prices, label = "Wiener Process") #Generalized Wiener Process a = 0.3 b = 1.5 deltaT = 1/daysCount changes_g = a*deltaT + np.random.randn(daysCount - 1, 1)*np.sqrt(1/daysCount) * b prices_g = np.zeros(days.shape) prices_g[0] = 10 for index in range(changes_g.shape[0]): prices_g[index + 1] = prices_g[index] + changes_g[index][0] plt.plot(days, prices_g, label = 'Generalized Wiener process') plt.legend() plt.show() - Ito’s Lemma（伊藤引理）
伊藤引理带入股票价格过程的公式：
应用在forward contracts上面：
股票价格的对数正态分布
前面说了股票价格变动符合布朗运动：
dS = μS dt + σS dz
dz = ε√Δt
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