I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.

My initial situation is the following. Consider two SDEs: \begin{align} dX_t &= f(X_t) dt + \sigma(X_t) dW_t\\ dY_t &= f(Y_t) dt + \tilde\sigma(Y_t) dW_t, \end{align} where $W_t$ is the standard 1D-Brownian motion, the initial data for $Y$ and $X$ are the same and the drift $f$ and the diffusion coefficients $\sigma,\tilde\sigma$ have sufficiently nice properties. Now assume that (for some reason) we know that the measures corresponding to this processes are mutually absolutely continuous.

Question: What can I conclude about the diffusion coefficients $\sigma,\tilde\sigma?$. Do they have to be equal (maybe only up to some symmetries or something like that)?

I appreciate any hint or reference. Thanks in advance.

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.

My initial situation is the following. Consider two stochastic differential equations: \begin{align} dX_t &= f(X_t) dt + \sigma(X_t) dW_t\\ dY_t &= f(Y_t) dt + \tilde\sigma(Y_t) dW_t, \end{align} where $W_t$ is the standard 1D-Brownian motion, the initial data for $Y$ and $X$ are the same and the drift $f$ and the diffusion coefficients $\sigma,\tilde\sigma$ have sufficiently nice properties. Now assume that (for some reason) we know that the measures corresponding to this processes are mutually absolutely continuous.

Question: What can I conclude about the diffusion coefficients $\sigma,\tilde\sigma?$. Do they have to be equal (maybe only up to some symmetries or something like that)?

I appreciate any hint or reference. Thanks in advance.