Aspects of Stochastic Integration beyond Standard Assumptions

This thesis deals with the theory of stochastic integration and tries to generalize some results beyond standard assumptions. One crucial part of defining a stochastic integral is the step to define it for martingales. If one follows a functional analytic approach introduced by Philip E. Protter, an inequality due to Burkholder is the necessary ingredient. This thesis generalizes this inequality to a setting in which stochastic processes with values in certain Banach spaces are considered, in particular when the integrand is Banach space-valued and the integrator real-valued. For such a Banach space we can define a stochastic integral of all càglàd processes with values in that Banach space against a real-valued semimartingale. Another major part of this thesis focuses on the Bichteler-Dellacherie theorem which is the characterization of stochastic processes which can be taken as integrator in the real-valued case. It tells that a stochastic integral can be defined precisely for càdlàg semimartingales and that those decompose into a local martingale and a finite variation process. We can extend this result by dropping the càdlàg assumption on the paths, using the analogue assumptions of the theorem and still decomposing the process on a level of versions. In particular we can find a càdlàg local martingale and a finite variation process such that for each time point the original process is the sum of those two, almost surely. The same ideas can be applied to the Doob-Meyer decomposition and we can again generalize this to supermartingales without continuity assumptions on its paths.