Properties of the local density process for general semimartingales (Q2784979)

Let the process \(\xi= (\xi_t)_{t\in R_+}\) be a semimartingale on the stochastic basis \((\Omega, F,{\mathbf F}= (F_t)_{t\in R_+},{\mathbf P})\) and on the stochastic basis \((\Omega, F,{\mathbf F},\widetilde{\mathbf P})\), where the probability measure \(\widetilde{\mathbf P}\) on \((\Omega, F)\) is such that \(\widetilde{\mathbf P}\overset{\text{loc}}\ll{\mathbf P}\). Here, the author writes \(\widetilde{\mathbf P}\overset{\text{loc}}\ll{\mathbf P}\Leftrightarrow \widetilde{\mathbf P}^t\ll{\mathbf P}^t\), where \({\mathbf P}^t={\mathbf P}|_{F_t}\), \(\widetilde{\mathbf P}^t= \widetilde{\mathbf P}|_{F_t}\). There exists a unique process \(z= (z_t)\in M_{\text{loc}}({\mathbf F},P)\) such that \(z_t= d\widetilde{\mathbf P}^t/d{\mathbf P}^t\), \(\forall t\in R_+\). Conditions under which the process \(\Lambda= \ln z\) is either a semimartingale or a special semimartingale are obtained and the corresponding decompositions are presented. The canonical representation for the semimartingale \(\Lambda\) and the corresponding triplet of predictable characteristics are derived. Also, the author proves the law of large numbers for \(\Lambda_t\), as \(t\to\infty\), and proves the theorem on weak convergence of \(\Lambda_t\). Applications of these results for the investigation of the asymptotical behavior of Neyman-Pearson tests are presented.