[size=12.000000pt]看讲义说A Brownian motion under measure P is a martingale under measure P (nodrift under measure P). 这句话大体意思我懂。EsP(Bt)=EP(Bs +(Bt−Bs))=EP(Bs)+EP(Bt−Bs)=Bs 但是这个under measure P是什么意思。在Brownian Motion的假设里貌似也没出现过under measure P啊。最近学mathematic finance学的云里雾里，还求问各位大神介绍有没有入门的书，有很详细的解释一步步的推出这些东西来。老师讲得什么martingale，Ito' lemma没有什么步骤推出来，完全跟不上。 NEED HELP !!!!!UGRENT!!! TKS A LOT!!!
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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.
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