On the loss of the semimartingale property at the hitting time of a level

TL;DR

This paper characterizes when the process g(Y) loses the semimartingale property at the hitting time of a level for a diffusion Y, providing deterministic conditions and applications in stochastic process theory.

Contribution

It introduces a precise classification of non-semimartingales of the first and second kind at hitting times, with a deterministic criterion for this loss based on g and Y's coefficients.

Findings

g(Y) can fail to be a semimartingale in two specific ways

Provides a deterministic if and only if condition for loss of semimartingale property

Constructs examples of diffusions that are semimartingales up to a stopping time but not afterwards

Abstract

This paper studies the loss of the semimartingale property of the process $g(Y)$ at the time a one-dimensional diffusion $Y$ hits a level, where $g$ is a difference of two convex functions. We show that the process $g(Y)$ can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the \textit{first} and \textit{second kind}. We give a deterministic if and only if condition (in terms of $g$ and the coefficients of $Y$) for $g(Y)$ to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion $Y$ on $[0,\infty)$ and a \emph{predictable} finite stopping time $\zeta$, such that $Y$ is a semimartingale on the stochastic interval $[0,\zeta)$, continuous at $\zeta$ and constant after $\zeta$, but is \emph{not} a semimartingale on $[0,\infty)$.