# Semimartingales and Shrinkage of Filtration

## TL;DR

This paper investigates how the semimartingale characteristics of a process change when considering a smaller filtration, focusing on both adapted and non-adapted cases, with implications for filtration shrinkage in stochastic processes.

## Contribution

It provides formulas for the semimartingale characteristics of processes under filtration shrinkage, extending understanding of optional projections and their characteristics.

## Key findings

- Derived formulas for $$-semimartingale characteristics when $X$ is $$-adapted.
- Established methods to compute $$-semimartingale characteristics of optional projections.
- Analyzed the relationship between $$- and $g$-semimartingale characteristics in filtration shrinkage.

## Abstract

We consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, 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 special $\mathbb{G}$-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 not $\mathbb{F}$-adapted, given that the $\mathbb{F}$-optional projection of $X$ is a special semimartingale, compute the $\mathbb{F}$-semimartingale characteristics of $\mathbb{F}$-optional projection of $X$ in terms of the $\mathbb{G}$-canonical decomposition and $\mathbb{G}$-semimartingale characteristics of $X$.