Sorry if my question is a bit basic. I am considering the default model as used eg in Vasicek (I think this goes back at least to Merton, though) that looks at an unobserved quantity modeling the assets of a counterparty through a logarithmic Wiener process $$dA_t = \mu A_tdt + \sigma A_tdB_t$$ with solution $$A_t = A_0e^{\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma\sqrt{t}B_t}.$$ Here, $B_t$ is a Brownian motion. I have see stated that the probability of default of the obligor over a certain time horizon $T$ is the probability that the assets of the obligor are below a certain threshold $D$ after time $T$: $$PD = P\left[A_T<D\right]\ .$$ This has the advantage of being readily computable, but conceptually shouldn't one consider the counterparty defaulted if their assets dip below $D$ at any time between $0$ and $T$?
Namely, I would define a stopping time by $$\tau:=\inf\left\{t\ge0:A_t\le D\right\}$$ and then define $$PD=P\left[\tau < T\right]\ .$$ Why is the first modeling of defaults used instead of the second one? What changes if we use this second definition? Do we still get a closed for solution for the PD (I haven't tried to work that out yet)?
NOTE: From a regulatory perspective for loans, even better would be to consider default if the counterparty's assets dip below a certain threshold and stay below it for at least a given period of consecutive time.
