I'm interested to finding a more analytical solution for one of the problems in Richard Bass's book Stochastic Processes.
Let $B_t$ be an one-dimensional Brownian motion. I'm asked to show that there exists some $\gamma$ such that if $t<\gamma$, then $$P(0\leq B_t \leq \delta/2) \geq \frac{1}{4}$$ and $$P(-\delta/2 \leq B_t \leq 0) \geq \frac{1}{4}.$$
For me it's easy to argue that, due to Gaussian distributions being symmetrical and 1-dimensional Brownian motion having the property $B_t\sim N(0,t)$, the problem is equivalent to estimating $P(B_t \leq |\delta/2|)$. By choosing $\gamma=\delta^2/4$, we are estimating the probability that a normal random variable is within one standard deviation from its mean, and from which we can show that such $\gamma$ exists.
However, I'm left with this lingering feeling that the problem could be solved with more finessé by using BM's theoretical properties. Any thoughts?
Here joint distribution of Brownian motion is mentioned nowhere, so this fact would hold for any stochastic process $X$ whose one-dimensional distributions are the same as for the Brownian motion. For example, $X$ can be a white noise with $0$ mean and variance linearly growing in time. As a result, no BM theory is really needed here beyond the fact that $X_t\sim \mathcal N(0, t)$. So your solution is the one that fits the problem the best way