I'm interested in the stochastic process followed by the marginal default density $e^{-\int_0^th(s)ds}h(t)$ in the case where the default intensity $h(t)$ follows a log-normal process.
Assuming
$$\frac{dh(t)}{h(t)} = \mu(t)dt+\sigma(t)dB(t)$$
an application of Ito's product rule shows that
$$ d\left(e^{-\int_0^th(s)ds}h(t)\right)=e^{-\int_0^th(s)ds}h(t)\left((\mu(t)-h(t))dt+\sigma(t)dB(t)\right) $$
I have generated some simple numerical examples which give the impression that the resulting process is not too far from log-normal (with shifted mean and volatility). But I'm interested in what might be known about the process more generally, and how far it diverges from log-normality. I imagine this must be a well-studied problem, and would be grateful if someone could direct me to where it has been studied in the literature.
