Ito diffusion density function

Question

As we all know, the solution to the geometric Brownian motion $$dX_t = b(X_t)dt + \sigma(X_t)dB_t$$ is subject to Gaussian distribution, so it is easy to write its density function.

Are there any other examples of the solution to the stochastic differential equation that can write their explicit density too? (if the coefficients that are not linear would be better). Furthermore, is there any solution of stochastic differential equation that follows other distribution except Gaussian distribution?

Answer 1

Not sure what you mean by the Gaussian distribution, that would depend on the values of $b(\cdot)$ and $\sigma(\cdot)$.

Note that

• $dX_t = \mu X_t dt + \sigma X_t dB_t$ is log-normal, while
• $dX_t = \mu dt + \sigma dB_t$ is normal (or Gaussian, as you mentioned), and
• if $\sigma = 0$ then $dX_t = \mu dt$ is deterministic altogether.

There are other classes of analytically solvable SDEs, for example the Ornstein-Uhlenbeck process.