Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

13Girsanov's theorem

13.1 Introduction through an example

Consider a complete probability space and a Wiener process defined on it. Consider an autonomous SDE

(13.1)

or, in the equivalent integral form,

(13.2)

with and satisfying a Lipschitz condition. Since the SDE is autonomous, this implies the existence and uniqueness of the solution.

The solution is a homogeneous diffusion process with drift coefficient and diffusion coefficient and the solution is in . Since we are considering a deterministic initial condition, automatically independent of the Wiener process, we can work with the ...