Blow up limits for SDE Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \, , \, X_0 = 0$$
with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous.
For each $c > 0$, define the process $Y^c$ on $[0, 1]$ by
$$Y^c_t := c^{-1/2} X_{ct}$$
Question: Is it true that as $c \to 0$, $Y^c$ converges in law to $\sigma(0) B_t$ for a Brownian motion $B$?
 A: First for the case of $\sigma\neq 0$ to get convergence as paths. As described here https://almostsuremath.com/2010/05/17/sdes-under-changes-of-time-and-measure/,
the above process is a time-changed Brownian motion
$$X_{t}=\beta_{A_{t}}$$
with $A_{t}:=\inf\{s\geq 0: \int_{0}^{s}\sigma(\beta_{s})^{-2}ds=t\}$. So if we start with $\frac{1}{\sqrt{c}}X_{ct}=\frac{1}{\sqrt{c}}\beta_{A_{ct}}$ and use the scaling law for Brownian motion, we get
$$\frac{1}{\sqrt{c}}X_{ct}\stackrel{d}=\beta_{\tilde{A}_{t}}$$
for $\tilde{A}_{t}:=\inf\{s\geq 0: \int_{0}^{s}\sigma(\sqrt{c}\beta_{s})^{-2}ds=t\}$. From here we use the continuity of $\sigma$ as $c\to 0$ to get convergence for the paths
$$\frac{1}{\sqrt{c}}\beta_{A_{ct}}\to \beta_{\sigma(0)^{2}t}\stackrel{d}{=}\sigma(0)\beta_{t}.$$
Case of sigma having zeroes
Stability approach
In the case of $\sigma$ having zeroes, one possible route is to use an appproximation and stability of SDEs since at the LHS and RHS there are no issues with $\sigma$ having zeroes. Let $\tilde{\sigma}(x):=\sigma(x)\vee \epsilon$ for $\epsilon>0$ and $X^{\epsilon}_{t}$ the corresponding SDE solutions.Then using a stability of SDEs eg. 6.9 Theorem in "Limit Theorems for Stochastic Processes" ( and ["Stability of strong solutions of stochastic differential equations"])2, we have the path convergence
$$P[\{X_{t}\}_{t}\in A]=\lim_{\epsilon \to 0}P[\{X_{t}^{\epsilon}\}_{t}\in A]=\lim_{\epsilon \to 0}P[\{\sigma(0)\vee \epsilon \beta_{t}\}_{t}\in A]=P[\{\sigma(0)\beta_{t}\}_{t}\in A]. $$
As mentioned in the comments of https://almostsuremath.com/2010/05/17/sdes-under-changes-of-time-and-measure/, the above situation is even true for weaker regularity for $\sigma$ satisfying $\sigma^{-2}$ being locally integrable (however, then one must be careful for uniqueness). For more details see here too
Characterization of martingale diffusions ending in $\{-1,1\}$
Engelbert & Schmidt time-change approach
Also in Shreves-Karatzas section 5.5, they go into a lot of detail on the existence/uniqueness of these Martingale-diffusions and their time-changes.
So here in 5.4 Theorem they show that the time change method is valid even for continuous $\sigma$ with zeroes.

