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.

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 $X$.