Let's consider two diffusion process $$dX_t = b(X_t) \, dt+\sigma(X_t) \, dW_t $$ and $$dX_t = c(X_t) \, dt+\theta(X_t) \, dW_t $$
with different drifts and diffusion coefficients. I would like to compute the Radon–Nikodym derivative of one of the process with respect to the other. When the diffusion coefficient are the same, this is related to the Girsanov theorem and I know how to do it.
When the diffusion coefficient are different, I have read in various place that the two process are not absolutely continuous and then the Radon–Nikodym derivative does not exist but I have not found any more details or inside about this point.
The question is then two-fold:
- What are the conditions on the diffusion coefficients such that the probability measure of the trajectories of one of the process is absolutely continuous with respect to the one one?
- In the case where there are absolutely continuous, what is the Radon–Nikodym derivative of the probability measure of the trajectories?