Enlargement of filtrations with random times for processes with jumps
The main purpose of the present paper is to extend a result due to \textit{J. Jacod} [Lect. Notes Math. 1118, 15--35 (1985; Zbl 0568.60049)] on the initial enlargement of filtrations with respect to random times. In Jacod's theorem it is assumed that the conditional distribution of the random time is absolutely continuous with respect to a nonrandom measure. Motivated by examples in the theory of insider trading the authors obtain extensions of Jacod's theorem where the reference measure mentioned above can be random. In the desired extension, the decomposition of the semimartingale (in the enlarged filtration) contains an extra term. An example concerns an insider who knows the time of the nth jump of the stock price of size bigger than a certain number. Another example is concerning the random time of the last jump before \(T\) of absolute size bigger than a given number (which is not a stopping time). These situations are not covered by Jacod's theorem.
scientific article; zbMATH DE number 5315299