It is frequent to see problems in stochastic calculus where we have to play with stopping times based on Brownian motions and a constant barrier $a>0$.
For example , $\tau= \inf\left\{t \geq0| W_t>a\right\}$ with $W$ a standard Brownian motion. We know that $P(\tau<\infty)=1$ because $limsup_{x \to \infty}{W_t}=\infty . $
We can increase the difficulty by using a linear or parabolic function instead of the constant $a$. For example , $\tau=\inf\{{t \geq0| W_t>\mu t+a\}}$ or $\tau=\inf\{{t \geq0| W_t>\beta t^2+\mu t+a\}}$ where $\mu, \beta$ are constant, and $a>0$ .
Here again , I managed to prove the cases where we have $P(\tau<\infty)=1$ , by using $\lim_{t\to \infty}{\frac{W_t}{t}=0}$
Indeed , if $\beta<0$, we have that $\lim_{t \to \infty}\frac{\beta t^2+\mu t+a}{t}<0$, same conclusion when $\beta=0$ and $\mu<0$.
If $\beta=0$ and $\mu>0$, reductio ad absurdum, assuming $P(\tau<\infty)=1$, we can build an increasing sequence of stopping time $\{\tau_k\}$, where the kth represents the kth time that the process crosses the polynomials and is strictly above it. They are almost surely finite by assumption.
However,$\frac{W_{\tau_k}}{\tau_{k}}>\mu>0$ and the stopping time sequence tends to $\infty$, we get an absurd result. The case where $\beta>0$ is treated the same way.
Now my curiousity brings me to the following question :
What happens to $P(\tau<\infty)=1$ when we use a polynomial of degree n ?
Let $\tau$ denotes the following stopping time $\tau=inf\{{t \geq0| W_t>\sum_{k=0}^{n}{a_kt^k}}\}$ with $\{a_k\}_{k=0}^{n}$ constant and $n\geq3$. What are the necessarry and sufficient conditions on $\{a_k\}_{k=0}^{n}$ in order to have $P(\tau<\infty)=1$?
I have always wondered whether that problem was solved, as I have never encountered it. Is there any general solution for that problem ?
Thank you.