Hello Quant Finance StackExchange,
Is there a simple way to find the crossing probabilities of a moving barrier, namely a barrier written in the form $U(t)=\alpha_1t^2 + \beta_1t + \gamma_1$ and $L(t)=\alpha_2t^2 + \beta_2t + \gamma_2$. (or if no solution, reduce the complexity to linear polynomial $\alpha t + \beta$.
I understand that there's a simple solution if the barrier is fixed, $U(t) = U$ and $L(t) = L$ and that $P(U$before$L)$, written here as $H(x)$, for a process $X_t$is given by
where $x$, $L\leq x \leq U$ is the starting position and $s(x)$ is the scaling function such that $Y_t = s(X_t)$ is a martingale. Example, for GBM with $2\mu \neq\sigma^2$,
Now, if I were to apply optional stopping for a moving barrier as I did a fixed barrier, it would seem that I'll get something like,
which would make it a random variable as the first exit time $\tau$ is random. Would getting the crossing probability as simple as taking the expectation, namely
Is this method reasonably logical?
(My rusty notation comes from me being two years removed from doing stochastic calculus at college. A rough explanation would do. Yes, I know there's tons of mathematical intricacies I conveniently overlooked.)