How to approximate stochastic volatility model with finite-state Markov chain? A common practice when computing solutions to stochastic dynamic optimization problems is to approximate an exogenous forcing process $z_{t+1} = \rho z_t + \sigma \epsilon_{t+1}$ with a finite-state Markov chain, e.g. by Tauchen's or Rouwenhorst's method. What would be a good way to discretize AR(1) process with stochastic volatility?

That is, if the original AR(1)-SV process looks something like this:
$$
\begin{split}
z_{t+1} &amp;= \rho_z z_t + \mathrm{e}^{v_t} \sigma_z \epsilon_{t+1} \\
v_{t+1} &amp;= \rho_v v_t + \sigma_v \eta_{t+1} 
\end{split}
$$
with $\epsilon, \eta$ being independent standard gaussian shocks, my goal is to obtain values $z_i$ and transition probabilities $p_{ij}$ ($i,j=1,\dots,n$) such that the corresponding Markov chain approximates the original continuous-valued process.
You could do a binomial-tree approximation to the process for $z_t$ and then have a different process control the number of steps you take on the tree. This preserves recombining property and it is essentially the method explored in On the Computation of  Continuous  Time Option Prices Using Discrete Approximations (Amin (1991)) 


  We  develop  a class of  discrete, path-independent  models to 
  compute prices  of American options  within the Black-Scholes  (1973)
  framework,  including  models in which state variables have 
  time-varying volatility  functions and models with  multiple  state
  variables. Time-varying volatility  functions are illustrated with 
  applications  to term structure models developed by  Vasicek  (1977) 
  and  Heath, Jarrow,  and Morton  (1988), (1990).  Distinct from 
  previous  work in the  literature,  the multivariate models suggested
  in this  paper are consistent with  arbitrarily large, though
  constant,  covariance functions. Finally,  we compare  and contrast
  the numerical  accuracy  of a  large  number of models with simulation
  results.


You could then use a binomial AR(1) process for the number of steps.