Let $X$ be a diffusion that is obtained from solving a SDE that is driven by a Brownian motion $B$. If certain well known conditions satisfied we can even obtain a strong solution, and it will be adapted to the filtration generated by $B$. Now, consider a SDE $$U_t = a(U_t)dX_t + b(U_t)dt.$$ Now, of course if certain conditions are met, we can obtain a strong solution to this SDE and since we can convert this to a SDE that is driven by $B$, filtration generated by $U$ will be a family of sub-sigma algebra of the sigma algebra generated by $B$. However, does it follow that strong solution $U$ to such SDE also adapted to the filtration generated by $X$? or, are there any well known conditions that ensures this?
2026-04-06 15:01:15.1775487675
Filtration generated by solutions of SDE driven by a diffusion
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