Change of Measure
Definition
Starting from a probability space {% (\Omega, \Sigma, \mu) %}, a change of measure is a change in the function {% \mu %} that assigns probability to the sample space.
Change of Measure for a Normal Distribution
Starting from the standard normal Distribution
{% \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} %}
The factor {% \frac{1}{2}x^2 %} in the exponent can be written as
{% \frac{1}{2}x^2 = - \frac{1}{2}(x - \mu)^2 - \mu x + \frac{1}{2} \mu^2 %}
This implies that the standard normal given above can be written as
{% \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} =
\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} (x - \mu)^2 } e^{-\mu x + \frac{1}{2} \mu ^2}
%}
{%
\displaystyle \mathbb{E}[h(X)] = \int_{- \infty}^{\infty} h(X) \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} =
\int_{- \infty}^{\infty} h(X) e^{-\mu x + \frac{1}{2} \mu ^2} \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} (x - \mu)^2 }
%}
Now defining
{% g(Y) = h(Y) e^{-\mu Y + \frac{1}{2} \mu ^2} %}
we have
{% \mathbb{E}_{\mathbb{P}}[h(X)] = \mathbb{E}_{\mathbb{\hat{P}}}[g(Y)] %}
where the probability under the measure {% \mathbb{P} %} is
{% \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2} %}
the probability under the measure {% \mathbb{\hat{P}} %} is
{% \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} (y - \mu)^2} %}