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Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law induced by $B_{1}(t)$ on $C[0,\infty)$. Here we follow the definitions of law is referred to

https://math.stackexchange.com/questions/90268/how-is-the-law-of-a-stochastic-process-defined/557519#557519

According to Girsanov theorem ( for example, P155. Thereom 8.6.3 in Fifth Edtion, Stochastic Differential Equations: An introduction with Application), there exists a Law $Q$ such that $B_{1}(t)$ is a standard Brownian motion under $Q$.

Is $Q$ equal to $P_{1}$ ?

I thought they are not equal to each other. The reason is that for fixed time $t$ the expectation of $B_{1}(t)$ under $Q$ is 0, and under $P_{1}$ its expectation should be $\mu t$. If my derivation is wrong, please point out where is my mistake.

If I am correct, a new question is how to compute $\frac{d P_{1}}{dP}$?

Recall that $\frac{d Q}{dP}$ is given by Girsanov theorem. Any references are very appreciated.

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No, that's the point of Girsanov's theorem. If $Q$ is equal to $P_1$, then nothing has changed. In order to make $B_1(t)$ a standard BM we need to transition to a new Law. Namely, $Q$.

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  • $\begingroup$ Your answer is very appreciated. I just added some more question. If $P_{1}\neq Q$, how to compute the radon nikodym derivative $\frac{d P_{1}}{d P}$? I am very confused with this computations. Some references said it can be obtained by Girsanov theorem. $\endgroup$ – user2781712 Nov 8 '13 at 0:06

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