For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say

$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$

Now if we consider an Ito integral, then

$$\left\vert\int_0^t f(s) \ dW(s)\right\vert \le \Vert f \Vert_{\infty} \vert \int_0^t \ dW(s)\vert$$

does not hold pointwise, but I was wondering whether this one holds probabilitically, i.e.

does there exist a constant $c(t)>0$ such that for all continuous $f$

$$\mathbb P\left(\vert W(t) \vert \Vert f \Vert_{\infty}\ge a\right)\ge c(t)\mathbb P\left(\left\vert\int_0^t f(s) \ dW(s)\right\vert \ge a\right)?$$

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say

$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$

Now if we consider an Ito integral, then

$$\left\vert\int_0^t f(s) \ dW(s)\right\vert \le \Vert f \Vert_{\infty} \vert \int_0^t \ dW(s)\vert$$

does not hold pointwise, but I was wondering whether this one holds probabilitically, i.e.

does there exist a constant $c(t)>0$ such that for all deterministic continuous $f$

$$\mathbb P\left(\vert W(t) \vert \Vert f \Vert_{\infty}\ge a\right)\ge c(t)\mathbb P\left(\left\vert\int_0^t f(s) \ dW(s)\right\vert \ge a\right)?$$