Before revision:  In the Gaussian case, those conditions have to do with the finiteness of the kurtosis and , for multinomial innovations, an inequality between the trend terms of the prices and of the volatility equations needs to be satisfied .
After revision:  In the Gaussian case, those conditions have to do with the finiteness of the price representation we are able to recover out of it the classical pricing formulas of Duan and Heston-Nandi, as well as hedging schemes that improve the performance of those proposed in the literature .