Mean Reverting Heston Model? Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity index returns, which I would argue show both volatility mean-reverting and asset-price mean reverting behavior. Clearly, where such an equity index could be thought to (at least approximately) mean-revert to some fixed long-term volatility, it also mean-reverts to the asset price point expected by annualized compounding returns (about 7% in real terms for the SP500).
As far as I know, the Heston Model does nothing to account for the second behavior. How would one modify the model to accommodate for that?
Edit: I tried to consider the following model. Any thoughts on it?
$dS_t = \sqrt{v_t}S_tdB_t^{(1)} + a_1(\mu_t - S_t)$
Where $u_t := \mathbb{E}[S_t]$ is the expected equilibrium at time t as per expected annualized compounding, $a_1$ is speed of mean reversion for price, $B_t^{(1)}$ is one dimensional Brownian Motion, and $v_t$ is the variance process $\{v_t, t\geq 0\}$ as defined with:
$dv_t = \sigma\sqrt{v_t}dB_t^{(2)} + a_2(v_t - \nu)$
Where, in turn, $\sigma$ is the constant vol of vol, $B_t^{(2)}$ is one dimensional Brownian Motion correlated to $B_t^{(1)}$ by $Cov(B_t^{(1)},B_t^{(2)})=\rho$, $a_2$ is speed of mean reversion for volatility, and $\nu$ is the long run average for volatility.
 A: In the Heston model, the underlying asset price $S_t$ follows a stochastic process similar to geometric Brownian motion, but with a stochastic volatility, $\sigma_t$, viz.
$$\tag{*}dS_t = \mu S_t \, dt + \sigma_t S_t \, dW_{S,t},$$
where variance $\sigma_t^2$ follows  a mean-reverting process
$$d \sigma_t^2 = \kappa(\theta - \sigma_t^2) \, dt + \beta \sigma_t\, dW_{\sigma,t}$$
Regardless of mean reversion in volatility, the process (*) will lead to some extremely unrealistic long-term behavior that is never observed with real equity indexes. For example, Monte Carlo simulation will generate many paths where (despite a positive drift) there can be a crash and no recovery for the duration of the simulation period.
A better alternative is a trending mean reversion model, such as
$$d (\log S_t - \mu t) = -\alpha(\log S_t- \mu t ) \, dt + \sigma_t dW_{S,t}, $$
where the log-price is pulled back to a stable trend at a rate that is proportional to the deviation from the trend. You could use
$\mu \approx 0.07$ for your purpose.
