Rough volatility models have become a prominent tool in quantitative finance due to their ability to cap- ture the rough nature of financial time series. However, these models typically have a non-Markovian structure, and this poses significant computational challenges. Existing methods for approximating these models often involve either complex quadrature techniques or require optimization algorithms op- erating in high-dimensional spaces, making them difficult and computationally intensive to implement in practice. This work introduces a novel and highly efficient quadrature rule for a Markovian approx- imation of rough volatility models, along with its theoretical error bounds. We verify the error bounds through a detailed error analysis and conduct several experiments on option pricing, demonstrating that the proposed method outperforms other state-of-the-art methodologies in terms of efficiency and accuracy across a range of roughness parameters. A further advantage of the approach is its ease to implement it in practice.

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets.
Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time stochastic processes are constructed as solutions to a certain class of fractional-order parabolic stochastic partial differential equations. Until now, the solution concepts, that have been formulated for these stochastic equations, either assume zero initial conditions for the process itself and (if existent) for all of its (mean-square temporal) derivatives, or set all (existing) higher-order derivatives at the starting time to zero. The aim of this thesis is twofold: Firstly, we generalize this class of stochastic processes in such a way that non-zero initial conditions (for the process and its temporal derivatives) can be meaningfully incorporated. Secondly, we show that for certain (integer) values of the parameter corresponding to the fractional power of the parabolic operator, the solutions satisfy certain Markov properties.
To this end, we first generalize the integration domain of stochastic Hilbert-space-valued Itô integrals to the whole real line. Afterwards, we present the modified spatiotemporal stochastic processes that facilitate incorporating nonzero initial conditions for the process and its derivatives at a given initial time. Lastly, we show that this proposed process satisfies a so-called multiple Markov property under certain conditions. Markov properties are desirable in numerical applications, because a Markov process allows for a much faster numerical approximation of its covariance operator.

In various scientific disciplines, measurement data is collected across space and over time with the aim of inferring information regarding an underlying stochastic spatiotemporal phenomenon. The high computational costs of current methods render this task intractable for the large datasets typically encountered in spatiotemporal statistics.

We propose a model based on Gaussian random fields, defined as the solutions to a class of space-time fractional stochastic partial differential equations (SPDEs) driven by Gaussian noise. The tunable parameters characterizing this class are expected to admit intuitive interpretations, in which case an efficient numerical approximation scheme for solutions to these SPDEs yields a powerful spatiotemporal model which is usable in practice.

The study of this class of SPDEs is the focus of this work. We define weak and mild solution concepts and show their equivalence under lenient conditions on the differential operators involved. Subsequently we establish results linking the well-posedness, spatiotemporal regularity and asymptotic covariance structure of the SPDEs to conditions on the model parameters, confirming their interpretability. Lastly, we specialize the SPDEs to a class of fractional ordinary differential equations in time, and describe a method to compute numerical approximations of their solutions. Future work is needed to analyze this scheme and generalize it to the original class of SPDEs.