Lower bound of cumulative probability of Brownian motion

Question

Let $B_t$ be the standard BM and $a>0$. I would like to estimate the lower bound of $\mathbb{P}(B_s-B_t\leq a)$. I think it can be given explicitly by $$ \mathbb{P}(B_s-B_t\leq a) = \dfrac{1}{\sqrt{2\pi(s-t)}} \int^a_{-\infty}\exp\left( -\dfrac{x^2}{2(s-t)} \right)dx.$$ But it seems that the Gaussian integral do not have a very explicit form. Do we have some good estimate for it in term of $a$, $s$ and $t$? Thank you so much!

Answer 1

Let $b:=a/\sqrt{s-t}$ (with $s>t$). Then $$ \mathsf{P}(B_s-B_t\le a)=\Phi(b)\ge 1-\frac{\phi(b)}{b}, $$ where $\Phi(\cdot)$ and $\phi(\cdot)$ are the standard normal cdf and pdf, respectively.