Let $X$ be the solution to the one dimensional SDE


$dX\_t = \mu(t, X\_t)dt + \sigma(t, X\_t) dW\_t$, for $t \in [0, T]$.


with $X\_0= x\_0$ a.s. for some $x\_0 \in \mathbb R$.


Here $W\_t$ denotes a standard Brownian motion, and we assume $\mu$ and $\sigma$ are Lipschitz continuous and uniformly bounded.


For every $\varepsilon > 0$, denote by $\mathcal S\_{\varepsilon}$ the event $\sup\_{t \in [0, T]} |W\_t| \leq \varepsilon$.


**Question:** Considering $X$ as a $C[0, T]$-valued random variable, is it true that the conditioned random variables $X| \mathcal S\_\varepsilon$ converge in law to the deterministic solution $Y\_t$ of


$dY\_t = \mu(t, Y\_t) dt$, with $Y\_0 = x\_0$ a.s.?



Conditioning an SDE on the event that the driving noise is small
The answer is yes, provided that you write your equation in Stratonovich form, rather than Itô form (and assuming that $\mu$ and $\sigma$ are sufficiently smooth in their arguments). The reason is that in one dimension the solution to the Stratonovich equation is a continuous map of $W$ in the sup-norm topology, [as observed by Doss in 1977](http://www.numdam.org/item/AIHPB_1977__13_2_99_0/).


This breaks in higher dimensions, but the answer to your question remains the same although I don't know if anyone wrote it up in precisely this way. (Various proofs of the Stroock-Varadhan support theorem use closely related variants of this statement. Note that it is again the Stratonovich formulation which is relevant.)


