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Aug 26, 2022 at 14:02 comment added Nawaf Bou-Rabee For a weak solution of the SDE you describe, it seems Girsanov might do the trick as implemented, e.g., mathoverflow.net/questions/424364/…. The idea is to relate the inhomogeneous SDE to a homogeneous one. See also mathoverflow.net/questions/426905/…
Jul 28, 2022 at 11:53 comment added GJC20 Thanks for the answer and apologies for the delayed reply due to family reasons. Indeed, for my case the drift term $B$ has the form $B(t,x)=b(x)+c(t)$ with $c\ge 0$. Thus, even $/sigma/equiv /sigma(x)$, the coefficients can not be homogeneous. I will read carefully the references that you listed and accept your answer as soon as I finish
Jul 24, 2022 at 14:20 history answered Nawaf Bou-Rabee CC BY-SA 4.0