It's a well known result due to Skorokhod that if for the SDE:
$$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$
the coefficients $b(t,x)$ and $\sigma(t,x)$ are assumed to be *continuous* and *bounded* then weak existence holds.
On Ikeda & Watanabe's book (in particular theorem IV.2.3.) the authors state:
> Given continuous coefficients $\sigma(x)$ and $b(x)$ of the (time
> homogeneous) markovian type SDE
>
> $$dX_t=b(X_t)dt+\sigma(X_t)dW_t$$ Then for any probability measure
> $\mu$ with compact support there exists a solution for the SDE whose
> initial distribution coincides with $\mu$.
(Worthy is to notice that the resulting solution may be explosive, but nonetheless this could be "solved" by assuming square integrability of the intial condition and linear growth.)
I am searching for further references regarding this particular result (which seems to be not so well-known). Neither Karatzas & Shreve nor Revuz & Yor seem to mention it. In the survey by Cherny and Engelbert they just mention the classic result due to Skorokhod and the same happens in Strook & Varadhan.
That's why I am curious about this result mentioned by Ikeda & Watanabe, which is less restrictive (even though it seems to apply just to the time homogeneous Markovian case).
Thanks in advance for any comment!
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