Definition
Change of Measure
Starting from a probability space {% (\Omega, \Sigma, \mu) %}, and given a random variable {% Z %} such that {% \mathbb{E}[Z] = 1 %}, we can define a new measure as
{% \displaystyle \mathbb{\hat{P}}(A) = \int_A Z(\omega) d P(\omega) %}
This implies that
{% \mathbb{\hat{E}}[X] = \mathbb{E}[XZ] %}
{% Z %} is called the Radon Nikodym derivative of {% \mathbb{\hat{P}} %} with respect to
{% \mathbb{P} %} and is written as
{% Z = \frac{\mathbb{\hat{P}}}{\mathbb{P}} %}
Radon Nikodym Theorem
Change of Measure for a Normal Variable
Under a probability measure {% \mathbb{P} %} such that {% X %} is distributed as a standard normal, then given the random variable
{% Z = e^{-\theta X - \frac{1}{2} \theta^2} %}
the random variable {% Y = X+\theta %}
is distributed under {% \mathbb{\hat{P}} %} with the standard normal distribution.
(see derivation)
Girsanov
Given a Brownian motion {% W(t) %} in a sample space {% (\Omega, \mathcal{F}, \mu) %}, where {% \mathcal{F}(t) %} is a filtration on this probability space, and given an adapted process {% \theta(t) %}, define the following
{% \displaystyle Z(t) = exp[-\int_0^t \theta(u) dW(u) - \frac{1}{2} \int_0^t \theta^2(u)du] %}
{% \displaystyle \hat{W}(t) = W(t) + \int_0 ^t \theta(u) du %}
If the following holds
{% \displaystyle \mathbb{E} \int_0^T \theta^2(u)Z^2(u)du < \infty %}
then the process {% \hat{W}(t) %} is a Brownian Motion under the measure {% \mathbb{\hat{P}} %}
defined as above.
see Shreve