Let $B$ be a standard Brownian motion in a filtered probability space. Assume, $K$ is bounded, adapted, continuous process and $\varepsilon > 0$. Let
$$ U_t = B_t - \int_{0}^{t}K_r ds, \quad t \geq 0 $$
Is there a choice for $K$ that ensures $U_t = \mathcal{O}(t^{1+\varepsilon})$ for $t \to \infty$?