I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE.
Would you study it empirically (like generating Monte-Carlo distribution and then performing statistical tests to fit it to known distros) or analytically? Examining the generated MC distribution of the model is a known option to me, however, do you know any common methods/approaches to study it analytically, too?
E.g., consider the SDE $$dr_t = a(b-r_t)\, dt + \sigma\sqrt{r_t}\, dW_t,$$
where $dW_t$ is a Brownian motion and $a,b, \sigma$ are constants. Then, $r_{t+T}$ is distributed as a Chi-squared distribution. How would you work to end up to such a conclusion?
Thank you in advance.
If I am right your process is a CIR process and is kind of tricky. Generally, to find a distribution of random variable you can solve its SDE, like in Black-Scholes model. From the wiki, you can interpret the CIR equation as a sum of squared Ornstein-Uhlenbeck which can be solved (pretty ugly) analytically. The random variable from an OU process will be normally distributed and you know that the sum of squared normal random variable follows a $\chi^{2}$ distribution. Therefore you can guess that $r_{t+T}$ will follow some kind of $\chi^{2}$ distribution.
You can also solve the Fokker-Planck equation of the process. Guess a functional form for the density, typically exponential, and apply Ito derivation. Then take the expectation and you will have an ODE. Solve it and you will get the probability distribution.