We consider a complete probability space , which is endowed with two filtrations, and , assumed to satisfy the usual conditions and such that . On this probability space we consider a real valued -semimartingale X.
The purpose of this work is to study the following two problems:
A. If X is -adapted, compute the -semimartingale characteristics of X in terms of the -semimartingale characteristics of X.
B. If X is a special -semimartingale but not -adapted, compute the -semimartingale characteristics of the -optional projection of X in terms of the -canonical decomposition and the -semimartingale characteristics of X.
In this paper problem B is solved under the assumption that the filtration is immersed in . Beyond the obvious mathematical interest, our study is motivated by important practical applications in areas such as finance and insurance (cf. Structured Dependence Between Stochastic Processes (2020) Cambridge Univ. Press).
"Semimartingales and shrinkage of filtration." Ann. Appl. Probab. 31 (3) 1376 - 1402, June 2021. https://doi.org/10.1214/20-AAP1621