In this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator is the generator of a strongly continuous semigroup on a Hilbert space ⁠, and the focus is on nonparabolic problems. The main results are optimal bounds for the uniform strong error...

Evolution equations are partial differential equations (PDEs) that describe evolution over time. To account for random perturbations, random coefficients or noise terms are added, often requiring a numerical solution. The contributions of this thesis are twofold. First, a joint convergence rate is presented for the approximation in randomness, space, and time using polynomial chaos for the random coefficients. Second, convergence rates for the pathwise uniform error in time are obtained for nonlinear stochastic PDEs in the hyperbolic Kato setting.

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error (Formula presented.) where p∈[2,∞), U is the mild solution, U^j is obtained from a time discretisation scheme, k is the step size, and N_k=T/k for final time T>0. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error (Formula presented.) which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

Ranking risks and countermeasures is one of the foremost goals of quantitative security analysis. One of the popular frameworks, used also in industrial practice, for this task are attack-defense trees. Standard quantitative analyses available for attack-defense trees can distinguish likely from unlikely vulnerabilities. We provide a tool that allows for easy synthesis and analysis of those models, also featuring probabilities, costs and time. Furthermore, it provides a variety of interfaces to existing model checkers and analysis tools.

Unfortunately, currently available tools rely on precise quantitative inputs (probabilities, timing, or costs of attacks), which are rarely available. Instead, only statistical, imprecise information is typically available, leaving us with probably approximately correct (PAC) estimates of the real quantities. As a part of our tool, we extend the standard analysis techniques so they can handle the PAC input and yield rigorous bounds on the imprecision and uncertainty of the final result of the analysis.

Cryptocurrencies such as Bitcoin have been one of the new major technologies of the last decade. In this paper, we assess the security of Bitcoin using attack-defense trees, an established formalism to evaluate the security of systems. In this paper, our main contributions are as follows: (1) We provide an extended attack-defense tree model for attacks on Bitcoin. (2) We demonstrate the general usability of existing analysis methods for attack-defense trees in this context. (3) We highlight further research directions necessary to extend attack-defense trees to a full-fledged overarching model for security assessment.