On the Loss of the Semimartingale Property at the Hitting Time of a Level

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 first and 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 predictable finite stopping time \(\zeta \) such that \(Y\) is a local semimartingale on the stochastic interval \([0,\zeta )\), continuous at \(\zeta \) and constant after \(\zeta \), but is not a semimartingale on \([0,\infty )\).