Given two SDE's, if the diffusion coefficients are pathwise uniformly close, can we say the same about the solutions to corresponding SDE's?

More precisely, consider $$ dX^1_t = b(X^1_t) dt + \sigma^1 \cdot \nu (X^1_t) dW $$ and $$ dX^2_t = b(X^2_t) dt + \sigma^2 \cdot \nu (X^2_t) dW $$ where $\sigma^1$ and $\sigma^2$ are continuous processes and $L$ is a fixed cadlag process.

Is it possible to make a statement along the following lines: For all $T \leq \infty$ and $\epsilon > 0$, there exists $\delta > 0$ such that $$ P(\sup_{0 \leq u \leq T} |\sigma^1_u - \sigma^2_u| > \delta) < \delta \Rightarrow P(\sup_{0 \leq u \leq T} |X^1_u - X^2_u| > \epsilon) < \epsilon? $$

(Assume integrability and growth conditions on $b, \nu, \sigma^1$ and $\sigma^2$ to ensure strong existence and uniqueness of $X^1$ and $X^2$.)

If so, does it make sense to extend the question to processes of the type $$ dX^i_t = b(X^i_t) dt + \sigma^i \cdot \nu (X^i_t) dW - dL, \, i = 1,2 $$ where $L$ is a fixed cadlag process?