A canonical setting and separating times for continuous local martingales (Q1016604)

The authors construct a certain canonical setting for continuous local martingales (CLMs) and find an explicit form of separating times [see \textit{A. S. Cherny} and \textit{M. A. Urusov}, Theory Probab. Appl. 48, No. 2, 337--347 (2003; Zbl 1070.60004), From stochastic calculus to mathematical finance. The Shiryaev Festschrift. Berlin: Springer. 125--168 (2006; Zbl 1103.60043)] for CLMs in this setting. Informally, a separating time for two measures \(\mathsf P\) and \(\widetilde{\mathsf P}\) on a filtered space \((\Omega,\mathcal F,(\mathcal F_t)_{t\geq0})\) is a stopping time \(S\) such that \(\mathsf P\) and \(\widetilde{\mathsf P}\) are equivalent before \(S\) and singular after it. Separating times can be used to establish criteria for absolute continuity and singularity between measures. The idea of the canonical setting for CLMs relies on the Dambis--Dubins--Schwarz theorem, which states that each CLM \(X\) starting from~0 is a time-changed Brownian motion. As canonical space for CLMs the authors consider the space \(\Omega^*\) of pairs \((\beta,\alpha)\) of a Brownian trajectory and a time-change trajectory. They prove that there is a one-to-one correspondence between distributions of CLMs and probability kernels from the Wiener space to the time-change path space; these kernels are non-anticipating in a certain sense. Given a non-anticipating kernel \(K\) one can define a probability measure \(\mathsf P\) on \(\Omega^*\) such that under \(\mathsf P\) the process \(\beta_{\alpha_.}\) is a ``canonical version'' on \(\Omega^*\) of CLMs corresponding to \(K\). The authors further discuss pure CLMs (PCLMs). They present several new definitions of PCLMs \(X\) (covering also the case \(\mathsf P(\langle X\rangle_\infty<\infty)>0\)), and show they are equivalent to definitions already existing in the literature. The separating time for measures \(\mathsf P\) and \(\widetilde{\mathsf P}\) on \(\Omega^*\) corresponding to non-anticipating kernels \(K\) and \(\widetilde K\) is shown to be ``the first time \(K\) and \(\widetilde K\) become singular'' under the condition that the time-change component \(\alpha\) is strictly increasing both under \(\mathsf P\) and under \(\widetilde{\mathsf P}\). Although it is natural to expect that such a result holds without the latter assumption, it is shown that this is not the case. For PCLMs the separating times take an especially simple form. Finally, criteria of absolute continuity and singularity for \(\mathsf P\) and \(\widetilde{\mathsf P}\) corresponding to PCLMs are established.