I was wondering about explicit examples of SDEs where strong and weak solutions are indeed different. I found online that the following is an example of and SDE where the two solutions are different:
$$ dX_t = |X_t|dt + X_tdW_t, \hskip 5pt X_0=1, \hskip 5pt W=\{W_t: t \ge 0\} \hskip 5pt BM, $$
but no details were provided. I can see that this is a modification of the usual SDE for a Geometric BM (writing $|X_t| = sgn(X_t)\cdot X_t$) but I cannot really see what steps I should perform. For a strong solution it would be enough to solve it, but the presence of $|X_t| $ is something I do not know how to handle (unless the fact that we start from $X_0=1 $ means that $X_t $ would never become negative as is the case for a Geometric BM)..
In any case, I cannot see the "easy way" to get the weak solution (i.e. what probability space should I choose?).
Any suggestion greatly appreciated. I do not think that this example should be complicated and it is also quite nice and it would be useful to have as an example of situation where strong and weak solutions differ.
Thank You