A note on stochastic integration with respect to optional semimartingales
Christoph Kühn, University of Frankfurt
Maximilian Stroh, University of Frankfurt
Abstract
In this note 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.
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