Multivariate Stochastic Volatility with Jumps and strong Tail Correlation

How to model two correlated stocks where the innovations are asymmetric and heavy-tailed? The model is estimated with MCMC on historical returns (not option data) and later used for VaR simulation. Consider the univariate SV model: $$ \begin{aligned} r_t &= \mu + e^{h_t}\epsilon_t \qquad \epsilon_t \sim N(0,1)\\ h_t &= \omega + \phi(h_{t-1}-\omega) + \bigl(J_t-\mathbb E[J_t]\bigr),\\ J_t &= \log \eta_t,\qquad \eta_t\sim \mathrm{InvGamma}(\nu/2,\nu/2),\\ \end{aligned} $$ For two assets, I want de