Perturbation of volatility term in an SDE

Question

Suppose $X, X^{\varepsilon}$, for $\varepsilon > 0$ are real valued stochastic processes satisfying the following SDE on $[0, T]$:

$dX = \mu(t, X_t) dt + \sigma (t, X_t) dW_t,$

$dX^{\varepsilon} = \mu(t, X^{\varepsilon}_t) dt + \sigma_{\varepsilon} (t, X^{\varepsilon}_t) dW_t,$

$X_0 = X^{\varepsilon}_0 = x_0$, for some $x_0 \in \mathbb R$.

with $\mu, \sigma, \sigma_{\varepsilon}$ uniformly bounded and Lipschitz continuous with uniform Lipschitz constant.

Suppose it is known that for every $\varepsilon > 0$, $\sigma(t, X_t) = \sigma_{\varepsilon} (t, X^{\varepsilon}_t)$ for all $t$ in a set of Lebesgue measure $T - \varepsilon$, where the set may depend on the random outcome $\omega$.

Question: Is it true that $X^{\varepsilon} \to X$ uniformly in $t$ in probability?

That is, $\lim_{\varepsilon \to 0+} \mathbb P(\sup_{t \in [0, T]} |X_t - X^{\varepsilon}_t| > \delta) = 0$ for all $\delta > 0$.

Answer 1

The techniques are similar to the answer here Small noise limits with irregular drift.

In the article "On stochastic differential equations with locally unbounded drift" they obtain a beautiful pointwise almost everywhere result for the coefficients that then gives uniform. So on that random set you have, we only need that eventually it converges to $\sigma$ almost everywhere.

The only caveat is that you need to assume uniform ellipticity for the volatility coefficient.

Then they get nice uniform convergence results.