Diffusion processes with different diffusion coefficients and absolute continuity

Question

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.

Answer 1

Well, if there are regions on the real line where the processes never get (which can be the case, e.g. if $f\equiv 1$ and $\sigma$ has zeroes with fast enough decay), then, clearly, you cannot say anything about the $\sigma$'s there.

Otherwise the quadratic variation is the invariant that ensures $\sigma=\pm\tilde{\sigma}$. Suppose that $|\sigma(x)| <a<b<|\tilde{\sigma}(x)|$ and let $\tau_x=\min\{t:X_t=x\}$. If $\tau_x<\infty$ with positive probability, then for a small $\epsilon>0$ also $|\sigma(X_t)| <a<b<|\tilde{\sigma}(X_t)|$ for $\tau_x<t<\tau_x+\epsilon$ with positive probability. But on this event $$\langle X_t\rangle_{[\tau_x,\tau_x+\epsilon]}=\int_{\tau_x}^{\tau_x+\epsilon}\sigma^2(X_t)dt<a^2\epsilon$$ and similarly $\langle X_t\rangle_{[\tau_x,\tau_x+\epsilon]}>b^2\epsilon$, which is a contradiction.