Let $(X_t)_t$ be a Markovian semi-martingale generating the filtration for the stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ on which a Brownian motion $(W_t)_t$ is defined. For every $\sigma>0$ consider the family of stochastic processes defined by the noisy process: $$ Y_t^{\sigma} = y_0 + \int_0^t f(X_s)ds \, + \sigma W_t. $$ Where $y_0\in \mathbb{R}$ and $f$ is non-constant and continuous.
Let $(\mathcal{F}_t^{\sigma})_t$ denote the filtration generated by $Y_t^{\sigma}$. Let $\{\sigma_n\}_n$ be a monotonically decreasing sequence in $(0,\infty)$ converging to $0$.

Under what conditions is $L^2(\mathcal{F}_t^{\sigma_n})\subseteq L^2(\mathcal{F}_t^{\sigma_{n+1}})$ and $\bigcup_{n} L^2(\mathcal{F}_t^{\sigma_n})=L^2(\mathcal{F}_t)$ for every $t>0$?


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