Volatility models versus intensity models: analogy and differences

We consider two popular classes of volatility models, the generalized autoregressive conditional heteroscedastic (GARCH) model and the stochastic volatility (SV) model. We compare these two models with two classes of intensity models, the integer-valued GARCH (INGARCH) model and the integer-valued stochastic volatility/intensity (INSV) model, which are corresponding integer-valued counterparts of the former. We reveal the analogy and differences of the models within the same class of volatility/intensity models, as well as between the two different classes of models.