I have a noisy signal that I filter. I want to know what is the probability that this signal goes over some threshold $a > 0$ in an interval $t\in [0,T] $.
In other words, I have some stochastic process $j(t) = \int_t^\infty f(\tau-t) dB(\tau)$, where $f(\tau)$ is some known filter function.
What is the probability that $j(t) > a$ for some $t\in [0,T] $ ?
I've tried to look at hitting times for Brownian motion, but I don't know how to apply the results to my (seemingly simple) problem.
