Semimartingales and shrinkage of filtration

Abstract

We consider a complete probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed to satisfy the usual conditions and such that $\mathbb{F}\subset \mathbb{G}$. On this probability space we consider a real valued $\mathbb{G}$-semimartingale X.

The purpose of this work is to study the following two problems:

A. If X is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of X in terms of the $\mathbb{G}$-semimartingale characteristics of X.

B. If X is a special $\mathbb{G}$-semimartingale but not $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of the $\mathbb{F}$-optional projection of X in terms of the $\mathbb{G}$-canonical decomposition and the $\mathbb{G}$-semimartingale characteristics of X.

In this paper problem B is solved under the assumption that the filtration $\mathbb{F}$ is immersed in $\mathbb{G}$. 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).