Pathwise closeness of solutions of SDE's

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?