求问A Brownian motion under measure P 是神马意思？

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关键词：Brownian Measure Motion Brown Under 神马

随机过程的教科书一般都有推导

fred_gs 发表于 2013-10-30 17:28
随机过程的教科书一般都有推导

能不能先帮我解释一下那个under measure P是什么意思。。。布朗运动哪里会体现这个概率P啊？谢谢！

I should say: W(t) is a Brownian motion under measure P, means under P measure, W(t) is a brownian motion.

Sounds I have explained nothing? But the implicit meaning is that, W(t) is NOT a brownian motion under other measures, e.g. risk neutral measure Q (if P refers to physical measure)

You will learn later that a Brownian motion under P will appear to have a drift term if the measure changes.
e.g. B(t)=a*t+w(t) is a not a standard Brownian motion under Q, but it can be a standard Brownian motion under P. Change of measure will kill the drift.

When we talk about stochastic variable or process, there is always a measure behind. You can take measure as a ruler that measures the probability. If the scale of the ruler changes, the probability changes. Under P, when you through a coin, 5:5 for head and tail, but there are other measures that can be 4:6 or 1:9. The important thing is not what that measure looks like but how do we change from one measure to another.

best,

P 是 measure, 定义probability,需要{Omega, F, P}。之后会Change Measure to Q, 如上面表示。