1. Agree, the time series model can be better viewed as "time-varying volatility" model. They have different roles. SV provides a distribution for pricing, while TS model is used to predict the volatility. But actually they have some relationship as I mentioned. See John Hull (Chapter 22 page 503 8ed) he gives a very simple example showing that GARCH(1,1) model is actually equivalent a stochastic process. Something like a lognormal Vasicek model.

2. Agree. But MC is not good for those models with non-normal density, such as CIR process.

3. As I know, Heston has a "closed" form solution in frequency domain, not in time domain. At last, still need numerical integral. What people usually do for SV model now is to use numerical method to find the joint density of the price and volatility. Then you can either use MLE to estimate the parameters or use numerical integral to do the pricing.

best,

我想到了一个有意思的问题：
给我股票数据，在常波动率假设下通过计量统计技术可以算出“波动率”；如果在随机波动率假设下，我希望能“逆向计算”出波动率的轨迹，或者退一步计算出和波动率有关的一条轨迹。如果波动率只是价格的函数，上述想法是可行的。如果波动率被另外的随机源驱动，得到的轨迹也许是某种“平滑”的结果。

Good thinking. This is probably another way to link the stochastic process with time series model.