Stochastic Integral

In the previous chapter, we saw that

\[ \exp \left( \int \sigma dW - \frac{1}{2}\int \sigma^2 dt \right) \]

is a strictly positive local martingale. On the other hand, we know that for a probability measure \(Q \sim P\), the process \(Z = (E[dQ/dP|\mathcal{F}_{t}])\) defines a strictly positive martingale. It turns out that these two aspects are related to each others through the concepts of stochastic exponential and the Girsanov Theorem.

• Stochastic Exponential
• Girsanov Transformationl