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 this report we will take a look at various proofs of Ramsey's theorem, some of the bounds that result from those proofs and applications of Ramsey's theorem. We will consider the proof of Ramsey himself, the proof of Skolem, the proof given by Erd\H{o}s and Szekeres and the proof of again Erd\H{o}s and Rado. The best upper bound for higher order Ramsey numbers is obtained by following the proof of Erd\H{o}s and Rado and this bound has only marginally improved since then. We will also state and prove the lower bound given by Erd\H{o}s and Hajnal. In the end we will apply Ramsey's theorem to both the Happy Ending problem and the monotone subsequence problem. The bounds we get for both problems using Ramsey's theorem, however, are quite weak compared to bounds that do not use Ramsey's theorem, so the theorem is more useful in proving the existence of a certain number than it is to find strong bounds for it.