Stability of compensator integrals and nonlinear operators under canonical p-th power variation convergence

Abstract

Let \(X\) and \((X^n)_{n\ge1}\) be c`adl`ag processes with predictable compensators \(ν\) and \(ν^n\). For \(p>1\), assume that \([X^n-X]^{(p)}_t\to0\) in probability, where \([\,\cdot\,]^{(p)}\) denotes canonical \(p\)-th power variation. We prove canonical \(p\)-th power-variation stability for a class of nonlinear compensator operators. Their atom responses combine a stable global Lipschitz cylinder component generated by finitely many compensator integrals with a nonlinear residual on a compact part of the mark space. A stopped compensation estimate and weighted cylinder approximation yield stability without assuming topological convergence of \(ν^n\) to \(ν\). Examples include covariance, cumulant, kernel and regularised transport responses. As a linear foundation, we prove stability of compensator integrals under local growth, locally uniform convergence of the integrands, jump continuity and barycentric tail control. The proof rests on two structural consequences of canonical \(p\)-th power-variation convergence: threshold isolation, which produces spatial bands eventually avoided by the limiting and approximating jump magnitudes, and compensator mass control, which uses predictable jump matching and localisation to construct an almost surely summable envelope for annular atom masses. These consequences require only the standing process and finiteness assumptions, independently of the additional hypotheses for compensator-integral stability. We also establish sufficient conditions for ucp stability of compensator integrals and, for \(p=2\), an equivalence criterion for ucp stability of the corresponding compensated jump integrals.