As we know, solution to $dX\_t=\mu dt+\sigma dW\_t$ is normal distributed and is light tailed; solution to $dX\_t=\mu X\_tdt+\sigma X\_t dW\_t$ is log-normal distributed and is heavy tailed. Is there any reference on discussion of criteria to identify the tail behavior from the driven SDE?



Tail of solutions of a stochastic differential equation
You want to consider:


$$dx\_t=f(x\_t)dt + \sigma(x\_t)dW\_t$$


In dimension 1, you can express analytically the stationary probability density (if it exists) as:


$$P(x)=\frac{1}{Z}\exp(-\int^x \frac{f(u)}{\sigma(u)^2} du)$$
where $Z$ is a normalizing constant.


Then you can express the various stationary moments using this formula and investigate the tail properties.


