We extend the recently developed rough path theory to the case of more rough noise and/or more singular Volterra kernels. It was already observed that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" developed, we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals.

This thesis is concerned with a solution theory for quasilinear singular stochastic partial differential equations. We approach the theory of regularity structures, a tool to tackle singular stochastic PDEs, from a new perspective which is well suited for, but not restricted to, quasilinear equations. In the first part of this thesis, we revisit the algebraic aspects of the theory of regularity structures. Although we approach regularity structures from a different perspective than originally done, we show that the same (Hopf-) algebraic structure is underlying. Trees do not play any role in our construction, hence the Hopf algebras underlying rough paths and regularity structures are not at our disposal. Instead, our alternative point of view gives a new (Lie-) geometric interpretation of the structure group, arising from simple actions on the nonlinearity of the equation and a parametrization of the solution manifold. In the second part of this thesis, we revisit the probabilistic aspects of the theory of regularity structures. We construct and stochastically estimate the centered model, which captures the local behaviour of the solution manifold. This is carried out under a spectral gap assumption on the driving noise, and based on a novel application of Malliavin calculus in regularity structures. In deriving the renormalized equation we are guided by symmetries, so that natural invariances of the model are built in. In the third part of this thesis, we make again use of the Malliavin derivative to obtain a robust characterization of the model, which persists for rough noise even as a mollification is removed. This allows for a simple derivation of invariances of the model that are not present at the level of approximations. Furthermore, we give a convergence result of models, which together with the characterization establishes a universality result in the class of noise ensembles satisfying uniformly a spectral gap assumption.

We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber to obtain a priori bounds for quasi-linear SPDEs. This approach replaces the index set of trees, used in the original constructions of Hairer et. al., by multi-indices describing products of derivatives of the corresponding nonlinearity. The two tasks of this thesis are: - Construction and estimates of the model. We first provide the construction of a model in the regular, deterministic setting, where negative renormalization can be avoided. We later extend these ideas to the singular case, incorporating BPHZ-renormalization under spectral gap assumptions as a convenient input for an automated proof of the stochastic estimates of the singular model in the full subcritical regime. - Characterization of the algebraic structures generated by the multi-index setting. We consider natural actions on functionals of the nonlinearity and build a (pre-)Lie algebra from them. We use this as the starting point of an algebraic path towards the structure group, which as in the regularity structures literature is based on a Hopf algebra. This approach further allows us to explore the relation between multi-indices and trees, which we express through pre-Lie and Hopf algebra morphisms, in certain semi-linear equations. All the results are based on a series of joint works with Otto, Tempelmayr and Tsatsoulis.

Diese Dissertation widmet sich der Untersuchung singulärer stochastischer partieller Differentialgleichungen (engl. SPDEs). Wir entwickeln Erweiterungen der bisherigen Lösungstheorien, zeigen fundamentale Beziehungen zwischen verschiedenen Ansätzen und präsentieren Anwendungen in der Finanzmathematik und der mathematischen Physik. Die Theorie parakontrollierter Systeme wird für diskrete Räume formuliert und eine schwache Universalität für das parabolische Anderson Modell bewiesen. Eine fundamentale Relation zwischen Hairer's modellierten Distributionen und Paraprodukten wird bewiesen: Wir zeigen das sich der Raum modellierter Distributionen durch Paraprodukte beschreiben lässt. Dieses Resultat verallgemeinert die Fourierbeschreibung von Hölderräumen mittels Littlewood-Paley Theorie. Schließlich wird die Existenz von Lösungen der stochastischen Schrödingergleichung auf dem ganzen Raum bewiesen und eine Anwendung Hairer's Theorie zur Preisermittlung von Optionen aufgezeigt. / This thesis is concerned with the study of singular stochastic partial differential equations (SPDEs). We develop extensions to existing solution theories, present fundamental interconnections between different approaches and give applications in financial mathematics and mathematical physics. The theory of paracontrolled distribution is formulated for discrete systems, which allows us to prove a weak universality result for the parabolic Anderson model. This thesis further shows a fundamental relation between Hairer's modelled distributions and paraproducts: The space of modelled distributions can be characterized completely by using paraproducts. This can be seen a generalization of the Fourier description of Hölder spaces. Finally, we prove the existence of solutions to the stochastic Schrödinger equation on the full space and provide an application of Hairer's theory to option pricing.