Local asymptotic quadraticity of stochastic process models based on stopping times (Q1893865)

A family of probability measures \((P_i : i \in I)\), \(I \subset \mathbb{R}^k\), on a filtered measurable space is considered. The filtration is assumed to be generated by a stochastic process which is a càdlàg \(\mathbb{R}^d\)-valued semimartingale with respect to each \(P_i\), its drift and jump characteristic depending on \(i\). The process is observed up to some stopping times \(\tau \to \infty\). Conditions are given which ensure the existence of good approximations of the log-likelihood process as \(\tau \to \infty\) (locally asymptotically quadratic with respect to variations of \(i\) and locally asymptotically normal or mixed normal).