I'm dealing with a problem in my thesis that involves proving the boundedness of a Stochastic Process. Here I present it in a really simplified version, hoping that it will inspire me in its more complicate version.
Let $\text{d}X_t= a \text{d}t +\text{d}W_t$ be a stochastic process defined for the times $t\ge0,$ such that $X_0=0$ and $a<0$. In other words, let $X_t$ be a drifted Brownian Motion $X_t=at+W_t$.
Prove that $$\mathbb{P}[\exists C>0: X_t<C \ \forall t\ge 0]=1.$$
In other words, I wish to show that the drifted Brownian motion is pathwise bounded from above, if the drift coefficient is negative.
What I've tried is to use hitting times to find the law of the running maximum of $W_t+at$, but it got quite messy and I didn't know how to go on. I also tried its discretized version but got stuck again.
Actually, I'm not even sure it's true, but experimentally it seems to be true. Here I attach a numeric experiment on a large scale, that shows behaviour that every other experiment shows, and suggests the boundedness from above.
Thanks a lot for the help!