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A note on stochastic integration with respect to optional semimartingales. (English) Zbl 1191.60068

Summary: We discuss the extension of the elementary stochastic Ito-integral w.r.t. an optional semimartingale. The paths of an optional semimartingale possess limits from the left and from the right, but may have double jumps. This leads to quite interesting phenomena in integration theory. We find a mathematically tractable domain of general integrands. The simple integrands are embedded into this domain. Then, we characterize the integral as the unique continuous and linear extension of the elementary integral and show completeness of the space of integrals. Thus, our integral possesses desirable properties to model dynamic trading gains in mathematical finance when security price processes follow optional semimartingales.

MSC:

60H05 Stochastic integrals
60G48 Generalizations of martingales
91G10 Portfolio theory
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