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Regular Integrals as Riemann Sum Limits

Quantum stochastic calculus on Boson Fock space was introduced by Hudson and Parthasarathy. A class of bounded integral processes, called regular semimartingale, was classified by Parthasarathy and Sinha and by Attal. Integrals, unbounded in general, of locally bounded processes using regular semimartingales as integrators can be defined. In this note, we study conditions for such integrals to become bounded adapted Riemann sum limits. These integrals present themselves as the price processes of quantum gain portfolios. Furthermore, they become martingales in a certain sense if the integrators are. This martingale property is the essential ingredient of a so-called Quantum Fundamental Asset Pricing Theorem.