Stochastic Exponentials and Logarithms on Stochastic Intervals -- A Survey

TL;DR

This survey explores stochastic exponentials and logarithms on stochastic intervals, detailing their definitions, properties, and interrelations for semimartingales, especially focusing on nonnegative local supermartingales and their inverses.

Contribution

It provides a comprehensive overview of stochastic exponentials and logarithms on stochastic intervals, highlighting their inverse relationship and properties for semimartingales.

Findings

Stochastic exponentials and logarithms are inverses for nonnegative local supermartingales.

Reciprocal of a stochastic exponential remains a stochastic exponential on a stochastic interval.

Definitions extend to semimartingales up to the first hitting time of zero.

Abstract

Stochastic exponentials are defined for semimartingales on stochastic intervals, and stochastic logarithms are defined for semimartingales, up to the first time the semimartingale hits zero continuously. In the case of (nonnegative) local supermartingales, these two stochastic transformations are inverse to each other. The reciprocal of a stochastic exponential on a stochastic interval is again a stochastic exponential on a stochastic interval.