Stability of Compensated Jump Integrals under Quadratic Variation Convergence

TL;DR

This paper proves that convergence of quadratic variation alone ensures the stability of compensated jump integrals for cadlag processes, without requiring traditional structural assumptions.

Contribution

It establishes ucp convergence of compensated jump integrals under quadratic variation convergence, introducing new structural mechanisms and removing the need for semimartingale or Markovian assumptions.

Findings

Quadratic variation convergence prevents jumps from crossing certain thresholds.

Compensator mass control mechanism ensures stability without global structural assumptions.

Results hold under local growth and uniform convergence conditions.

Abstract

We study the stability of compensated jump integrals under convergence of quadratic variation alone. Let \(X\) and \(\{X^n\}_{n\ge1}\) be c\`adl\`ag processes with jump measures \(\mu,\mu_n\) and predictable compensators \(\nu,\nu_n\). Under the assumption \[ [X^n-X]_t \to 0 \qquad\text{in probability}, \] we establish ucp convergence of compensated jump integrals of the form \[ \int_0^. \int_{\mathbb R} f_n(s,x)(\mu_n-\nu_n)(ds,dx) \] under local linear growth and locally uniform convergence assumptions on the integrands. The proof is based on two structural mechanisms. The first is a forbidden bands principle, showing that quadratic variation convergence prevents jumps from crossing suitably chosen moving threshold regions. The second is a compensator mass control mechanism, which combines threshold-separated alignment of large predictable jumps with a counting argument for the associated compensator atoms. The results require neither semimartingale convergence, convergence of characteristics, uniform tightness, nor global structural assumptions such as independence, stationarity, or Markovianity. More broadly, they show that quadratic variation convergence imposes a substantially stronger rigidity on the jump organization of c\`adl\`ag processes than one might initially expect.