# A bound for the occupation time of a diffusion

Let $$sigma: mathbb R times mathbb R to mathbb R$$ be a Lipschitz continuous function bounded below by some $$M > 0$$.

Let $$W$$ be a standard Brownian motion, and let $$X$$ be the solution to the SDE

$$dX_t = sigma(t, X_t) dW_t$$

with $$X_0 = 0$$.

Fix $$T > 0$$. Does there exist, for every $$varepsilon, h > 0$$ a $$delta > 0$$ such that

$$mathbb Pleft(int_{0}^T mathbf 1_{(-delta, delta)} (X_s) ds > hright) < varepsilon?$$