Mean reverting SDE applications

Question

I know some of famous mean reverting stochastic differential equations . such as below.Please help me to make this list better .If you know one or more of this kind (of equation) please add below .I am thankful for your help in advanced .

\begin{array}{c||c} Model- name & equation \\ \hline Vasicek- model (1977) & d{{r}_{t}}=(\theta -\alpha {{r}_{t}})dt+\sigma d{{W}_{t}} & \\ \hline Cox–Ingersoll–Ross- model (1985) & d{{r}_{t}}=(\theta -\alpha {{r}_{t}})dt+\sqrt{{{r}_{t}}}\sigma d{{W}_{t}} & \\ \hline Hull–White- model (1990) & d{{r}_{t}}=({{\theta }_{t}}-\alpha {{r}_{t}})dt+{{\sigma }_{t}}d{{W}_{t}} &\\ \hline Fokker–Planck -equation & \frac{{\partial f}}{{\partial t}} = \theta \frac{\partial }{{\partial x}}[(x - \mu )f] + \frac{{{\sigma ^2}}}{2}\frac{{{\partial ^2}f}}{{\partial {x^2}}}\\ \hline stochastic -version-of- falling -object & dv = (g - \frac{k}{m}v)dt - \frac{k}{m}\alpha {dW_t}\\ \hline Schwartz- 1997- commodity -models & \frac{dS}{S} = \alpha(\mu-lnS)\,dt +\sigma\, dW \\ \hline Schwartz -1997- commodity- models & \begin{aligned} \frac{dS}{S} &= (r-y)\,dt +\sigma\, dW \\ dy &= \alpha(\theta-y)\,dt+\epsilon \,dZ \\ dWdZ&=\rho\,dt \end{aligned}\\ \hline Heston- model & \begin{aligned} dS_t &= \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \\ d\nu_t &= \kappa(\theta - \nu_t)\,dt + \xi \sqrt{\nu_t}\,dW^{\nu}_t \end{aligned} \\ \hline \end{array}

Answer 1

Schwartz 1997 commodity models:

1 factor $$ \frac{dS}{S} = \alpha(\mu-lnS)\,dt +\sigma\, dW $$

2 factors $$ \begin{aligned} \frac{dS}{S} &= (r-y)\,dt +\sigma\, dW \\ dy &= \alpha(\theta-y)\,dt+\epsilon \,dZ \\ dWdZ&=\rho\,dt \end{aligned} $$

Heston model $$ \begin{aligned} dS_t &= \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \\ d\nu_t &= \kappa(\theta - \nu_t)\,dt + \xi \sqrt{\nu_t}\,dW^{\nu}_t \end{aligned} $$

Answer 2

You could list the Cheyette model which is particular case of the HJM model,

$$r_t=\phi_t+x_t$$ $$dx_t=(y_t-\kappa(t)x_t)dt+\eta(t)dW(t)$$ $$dy_t=(\eta(t)^2-2\kappa(t)y_t)dt $$

where $\phi_t$ and $\kappa$ are deterministic.

You would note that if $\eta$ is deterministic, the model is Gaussian. Better, you can choose $\eta$ in a way that the extended Vasicek model is a particular case of the Cheyette model.