Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ – thus we made a change of time. Let us denote an original measure as $P$. How to find measure $Q$ such that the process $Y$ is obtained through the process $X$ not with the change of time but with a change of measure, i.e.
$$
\mathcal{Law}(Y\text{ under }P) = \mathcal{Law}(X\text{ under } Q).
$$
Maybe it is possible to find the density process
$$
Z_t = \frac{dQ_t}{dP_t}
$$
like for the case when we're looking for the martingale measure? Unfortunately, now it seems the Girsanov theorem is useless for my case.
Probability – Change of Time or Change of Measure
measure-theorypr.probabilitystochastic-processes
Best Answer
for example, if $\sigma$ is a constant, then $Y$ satisfies $dY_t = \sqrt{a} \sigma dW_t$ so that the law $\mathbb{Q}_Y$ and $\mathbb{Q}_X$ of the processes $Y$ and $X$ on the Wiener space $C([0;T],\mathbb{R})$ are generaly singular: in other words $\frac{d \mathbb{Q}_Y}{d \mathbb{Q}_X} = 0$ if $|a| \neq 1$.