Semimartingales and Shrinkage of Filtration.

Summary We consider a complete probability space (Ω, F, P), which is endowed with two fitrations, G and F, assumed to satisfy the usual conditions and such that F ⊂ G. On this probability space we consider a real valued special G-semimartingale X. The results can be generalized to the case of R^n valued special semimartingales, in a straightforward manner. We fix a truncation function with respect to which the semimartingale characteristics are computed. The purpose of this work is to study the following two problems: A. If X is F-adapted, compute the F-semimartingale characteristics of X in terms of the G-semimartingale characteristics of X. B. If X is not F-adapted, given that the F-optional projection of X is a special semimartingale, compute the F-semimartingale characteristics of F-optional projection of X in terms of the G-canonical decomposition and G-semimartingale characteristics of X.