# Nested subspaces of measurable functions through noise

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 $$t>0$$?