Local asymptotic quadraticity of stochastic process models based on stopping times

Consider a semimartingale whose drift and jump characteristic depend on an unknown parameter. The process is observed up to some stopping time η. We establish conditions which ensure that the resulting statistical model admits locally a quadratic approximation of the log-likelihood process with asymptotics as η → ∞. This provides an important step in the solution of the inference problem for the unknown parameter based on random stopping.