# Conditioning an SDE on the event that the driving noise is small

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.?

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.

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.)

• Thank you! What is your gut feeling as to the whether the Ito form is true/false? I feel like because of the quadratic term in the Ito lemma (for Ito integrals) something might break due to roughness of the paths of $W_t$ - in the sense that controlling the sup norm of $W_t$ will not be sufficient. I don’t know how to push this idea further rigorously though… Sep 20, 2021 at 10:07
• The two statements are mutually exclusive... Sep 20, 2021 at 10:08
• Oh, I will try to work that out. Thanks! Sep 20, 2021 at 10:11

As an approximation for a counterexample, consider $$X_t=\sin(W_t+1)$$. It has the stochastic differential $$dX_t=(-(1/2)\sin(W_t+1))dt+\cos(W_t+1)dW_t$$ with initial condition $$X_0=\sin(1)$$. Then $$X|{\mathcal{S}_\varepsilon}$$ converges to a constant function, which is not the solution of the diffusion-less equation $$dX_t=(-(1/2)\sin(1))dt$$.

This equation is not exactly in your form, but it is if you allow $$2$$-dimensional equations and consider $$W$$ as the first coordinate.

• That’s a really nice idea. I’m also convinced its no in the one dimensional case but have yet to find a specific counterexample. Sep 20, 2021 at 8:27
• Hmm, though in your example $W$ and $X$ end up being very “correlated” whereas in the original problem $X$ is only linked to $W$ via the defining SDE. Maybe this changes things in a key way… Sep 20, 2021 at 8:29