On the Dynamics of Semimartingales with Two Reflecting Barriers

Abstract

We consider a semimartingaleXwhich is reflected at an upper barrierTand a lower barrierS, whereSandTare also semimartingales such thatTis bounded away fromS. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers toX,S, andT. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions onX,S, andT, e.g. if we takeX,S, andTto be independent martingales (which satisfy some extra technical conditions).