We will review Girsanov theorem, and related change of measure.

Given two probability measures and on and a filtration , they are also probability measures on , and we will use notation and when the measures are restriced by . Suppose , then , and write

**Lemma 1** * Suppose , then is a -martingale. *

*Proof:* For all , ,

**Proposition 2 (Bayes rule)** * Let be equivalent two probability measures on the probability space . Let be a filtration with . Then,*

*Proof:* By Lemma 1, is a martingale. It’s enough to show,

It follows by

**Lemma 3** * Under the same assumptions of Proposition 2, is a -martingale if and only if is -martingale. *

*Proof:* By Proposition 2 and Lemma 1, we have the following equivalent identities: For any ,

- ;
- ;
- ;
- ;

Hence, we conclude the result.

Indeed, Lemma 1 is a sepcial case of Lemma 3 with .

**Example 1** * Let be two independent standard Brownian motion on . Given*

Find all equivalent probability measure under which is a Brownian motion.

Since, , it’s enough to find all density function satisfying

- is a martingale with ;
- is a martingale with ;

Since is a martingale, it must takes form of

By Ito’s product formula, we also have

for some martingale . Thus, , and could be any. As a conclusion, is a equivalent probability measure under which is a Brownian motion, if and only if, is a solution of

for some satisfying integrability condition

**Theorem 4 (Girsanov)** * Let be a standard Brownian motion w.r.t . is exponential martingale with being of the form*

Then, a new process defined by

is a -Brownian motion.

*Proof:* First, we use Lemma 3 to show that is a centered -martingale. Then, by the fact , we can complete the proof using Levy’s characterization.

The following partial extension of Girsanov’s theorem will be of use in applications to finance. First recall that, if Levy-type stochastic integral be given by

then is a local martingale if and only if for ,

**Proposition 5** * Let be a local martingale of the form*

where . Let be an exponential martingale of (2). Then, the new process given by

is a local martingale, provided that the integral exists.

*Proof:* Use Lemma 3.

A sufficient condition for the existence of the integral in is that .

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