We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$. I would like to find the conditions when is it possible to do with a homogeneous Markov process in a continuous time? It seems to be true for the Markov chains, but it's not true for piecewise deterministic Markov processes (because of pure deterministic part between jumps).

Maybe we can consider at first the diffusion case. Let $$ dX_t = \mu(X_t)dt+\sigma(X_t)dw_t. $$

Is there a change of measure $P\to Q$ such that under the new measure $$ dX_t = a\mu(X_t)dt+\sigma(X_t)\sqrt{a}\cdot dw_t $$ where $a>0$?