The Ito Formula for Essentially Self-Adjoint Quantum Semimartingales

Abstract

The quantum Ito formula has so far been proved for regular (bounded) quantum semimartingales We give three different extensions to classes of essentially self-adjoint (unbounded) quantum semimartingales. The first extension is to quantum semimartingales whose integrands have chaos matrices controlled by certain scalar matrices The second extension is to perturbations of classical, possibly unbounded, quantum semimartingales by bounded quantum semimartingales The third extension is obtained by perturbing an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula by a regular quantum semimartingle.