Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

This paper offers an existence result for finite dimensional stochastic differential inclusions with maximal monotone drift and diffusion terms. Kravets studied only set-valued drifts in [5], whereas Motyl [4] additionally observed set-valued diffusions in an infinite dimensional context. In the proof we make use of the Yosida approximation of maximal monotone operators to achieve stochastic differential equations which are solvable by a theorem of Krylov and Rozovskij [7]. The selection property is verified with certain properties of the considered set-valued maps. Concerning Lipschitz continuous set-valued diffusion terms, uniqueness holds. At last two examples for application are given.

This paper offers an existence result for finite dimensional stochastic differential inclusions with maximal monotone drift and diffusion terms. Kravets studied only set-valued drifts in [5], whereas Motyl [4] additionally observed set-valued diffusions in an infinite dimensional context. In the proof we make use of the Yosida approximation of maximal monotone operators to achieve stochastic differential equations which are solvable by a theorem of Krylov and Rozovskij [7]. The selection property is verified with certain properties of the considered set-valued maps. Concerning Lipschitz continuous set-valued diffusion terms, uniqueness holds. At last two examples for application are given.

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

Diese Dissertation präsentiert neue Techniken der Integration für verschiedene Probleme der Finanzmathematik und einige Anwendungen in der Wahrscheinlichkeitstheorie. Zu Beginn entwickeln wir zwei Zugänge zur robusten stochastischen Integration. Der erste, ähnlich der Ito’schen Integration, basiert auf einer Topologie, erzeugt durch ein äußeres Maß, gegeben durch einen minimalen Superreplikationspreis. Der zweite gründet auf der Integrationtheorie für rauhe Pfade. Wir zeigen, dass das entsprechende Integral als Grenzwert von nicht antizipierenden Riemannsummen existiert und dass sich jedem "typischen Preispfad" ein rauher Pfad im Ito’schen Sinne zuordnen lässt. Für eindimensionale "typische Preispfade" wird sogar gezeigt, dass sie Hölder-stetige Lokalzeiten besitzen. Zudem erhalten wir Verallgemeinerungen von Föllmer’s pfadweiser Ito-Formel. Die Integrationstheorie für rauhe Pfade kann mit dem Konzept der kontrollierten Pfade und einer Topologie, welche die Information der Levy-Fläche enthält, entwickelt werden. Deshalb untersuchen wir hinreichende Bedingungen an die Kontrollstruktur für die Existenz der Levy-Fläche. Dies führt uns zur Untersuchung von Föllmer’s Ito-Formel aus der Sicht kontrollierter Pfade. Para-kontrollierte Distributionen, kürzlich von Gubinelli, Imkeller und Perkowski eingeführt, erweitern die Theorie rauher Pfade auf den Bereich von mehr-dimensionale Parameter. Wir verallgemeinern diesen Ansatz von Hölder’schen auf Besov-Räume, um rauhe Differentialgleichungen zu lösen, und wenden die Ergebnisse auf stochastische Differentialgleichungen an. Zum Schluß betrachten wir stark gekoppelte Systeme von stochastischen Vorwärts-Rückwärts-Differentialgleichungen (FBSDEs) und erweitern die Theorie der Existenz, Eindeutigkeit und Regularität der sogenannten Entkopplungsfelder auf Markovsche FBSDEs mit lokal Lipschitz-stetigen Koeffizienten. Als Anwendung wird das Skorokhodsche Einbettungsproblem für Gaußsche Prozesse mit nichtlinearem Drift gelöst. / In this thesis new robust integration techniques, which are suitable for various problems from stochastic analysis and mathematical finance, as well as some applications are presented. We begin with two different approaches to stochastic integration in robust financial mathematics. The first one is inspired by Ito’s integration and based on a certain topology induced by an outer measure corresponding to a minimal superhedging price. The second approach relies on the controlled rough path integral. We prove that this integral is the limit of non-anticipating Riemann sums and that every "typical price path" has an associated Ito rough path. For one-dimensional "typical price paths" it is further shown that they possess Hölder continuous local times. Additionally, we provide various generalizations of Föllmer’s pathwise Ito formula. Recalling that rough path theory can be developed using the concept of controlled paths and with a topology including the information of Levy’s area, sufficient conditions for the pathwise existence of Levy’s area are provided in terms of being controlled. This leads us to study Föllmer’s pathwise Ito formulas from the perspective of controlled paths. A multi-parameter extension to rough path theory is the paracontrolled distribution approach, recently introduced by Gubinelli, Imkeller and Perkowski. We generalize their approach from Hölder spaces to Besov spaces to solve rough differential equations. As an application we deal with stochastic differential equations driven by random functions. Finally, considering strongly coupled systems of forward and backward stochastic differential equations (FBSDEs), we extend the existence, uniqueness and regularity theory of so-called decoupling fields to Markovian FBSDEs with locally Lipschitz continuous coefficients. These results allow to solve the Skorokhod embedding problem for a class of Gaussian processes with non-linear drift.