In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes Monte Carlo simulations viable if not necessary even for the calculation of European options.
However, all models used in mathematical finance that come immediately to my mind are processes for which the distribution is known, such as geometric Brownian motion and the Cox-Ingersoll-Ross process.
What would be examples for diffusion processes, preferably widely used, of the above form for which the distribution of $X_T$ is unknown?