On the dynamics of semimartingales with two reflecting barriers (Q2854074)

The authors consider the following Skorokhod problem: For a semimartingale \(X\) with \(X(0)=0\) and semimartingales \(S,T\) reflecting \(X\) as an lower and upper bound find \((V,L,U)\) satisfying \(S(t) \leq V(t) \leq T(t)\) for all \(t\), \(L\) and \(U\) nonnegative, finite, nondecreasing and right-continuous, such that it holds: \(V(t)=V(0)+X(t)+L(t)-U(t)\). \(L\) and \(U\) are minimal in the sense that they only increase when \(V\) reaches the lower or upper barrier. For this problem the authors derive basic mechanisms to deal with this reflection and derive general properties. They construct a link between the process \(V\) and the local times to \(X\), \(S\) and \(T\) via stochastic integrals and show the existence of the solution \((V,L,U)\). Moreover, for a Lévy process \(X\) they establish an expression for \(\operatorname{E}[U(1)]\) in terms of the stationary measure of \(V\) and the Lévy measure of \(X\). For a martingale \(X\) and a downwards skip-free martingale \(T\) independent of \(X\), the structural relationship becomes more simple. Finally, the authors illustrate their results with some examples.