Optimal portfolio allocation with volatility and co-jump risk that Markowitz would like

We study a continuous time optimal portfolio allocation problem with volatility and co-jump risk, allowing prices, variances and covariances to jump simultaneously. Differently from the traditional approach, we deviate from affine models by specifying a flexible Wishart jump-diffusion for the co-precision (the inverse of the covariance matrix). The optimal portfolio weights that solve the dynamic programming problem are genuinely dynamic and proportional to the instantaneous co-precision, reconciling optimal dynamic allocation with the static Markowitz-type economic intuition. An application to the optimal allocation problem across hedge fund investment styles illustrates the importance of having jumps in volatility associated with jumps in price.