I am a non-mathematician trying to apply stochastic calculus. I have have a process $X_t$ taking values in $[0,\infty)$. It is defined by
\begin{equation} X_t=exp\left(\int_{0}^{t}\Theta(u)dW(u)\right), \end{equation}
where $\int_{0}^{t}\Theta(u)dW(u)$ is a stochastic integral. $\Theta_t$ is also stochastic but its values are determined by the Wiener process $W_t$. How can I show (or what assumptions do I need to impose on $\Theta_t$ to guarantee) that the process does not converge almost surely as $t\rightarrow \infty$ to $0$ (or any other number). Moreover, how to show that it does not approach infinity, for example that there is a bound $K$ such that
$Prob(X_t<K)>0$ $\forall t$.