# Tail of solutions of a stochastic differential equation

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?

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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.

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