Before revision:  Regarding this subject, we focus on GARCH time series models with Gaussian and multinomial innovations and we provide specific conditions under which those martingale measures are appropriate in the context of quadratic hedging.
After revision:  Regarding this subject, we focus on GARCH time series models with Gaussian and multinomial innovations and we provide specific sufficient conditions that have to do with the finiteness of the kurtosis, under which those martingale measures are appropriate in the context of quadratic hedging.