In this bachelor's thesis we will solve the Dirichlet problem with an L^p(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature.

In this research, the returns of four cryptocurrencies (Bitcoin, Litecoin, Ripple and Ethereum) were analyzed in order to answer the following research question: “How do the returns of Bitcoin and other altcoins behave over time, and what can we say about extreme values for losses and profits?” With respect to volatility, cryptocurrencies can still be considered extremely volatile. For Bitcoin, the least volatile of the four, we found an annual volatility of approximately 70% based on daily exchange rates. For Ethereum, the most volatile of all four, this percentage was closer to 130%. Also, several distributions were fitted on the returns. It is shown that the Generalized Hyperbolic Distribution is the best fit for all four cryptocurrencies, apart from the tails in some cases.
The tails were investigated seperately by using Extreme Value Analysis and by looking into both empirical and theoretical risk quantities (the Value at Risk and Expected Shortfall). Bitcoin appears to be the least risky of all four cryptocurrencies, but also the least profitable, whereas Ripple appears to be the most risky and also the most profitable.
Compared to previous research, Bitcoin has also become less risky, showing a less fat tail for the losses than before. For Litecoin and Ripple, the reverse is true, as they appear to have become riskier. For Ethereum, no comparisons could be made, as this is a relatively new cryptocurrency that has not been investigated much yet. When tested for Paretianity, the left tails of Litecoin and Ripple appear to Pareto distributed: the losses seem to exhibit heavy tail behavior. For the profits, the tails turned out to be even heavier and can therefore also be considered Paretian. These results were confirmed by Maximum to Sum ratio plots, indicating infinite third and fourth moments for the losses and profits of Litecoin and Ripple, but not for Bitcoin and Ethereum. The results have implications for investment and risk management purposes.

The aim of this thesis is to provide a formula for the value of a correlation swap. To get to this formula, a model from an article by Bossu is inspected and its resulting expression for fair the fair value of a correlation swap is simulated. The Jacobi process will be defined and two discretization schemes will be compared for it. Methods are discussed to make simulations of the Jacobi process as accurate as possible, making sure it crosses its boundaries -1 and 1 as little as possible. It will be shown that a correlation swap can be hedged by dynamically trading variance dispersion trades. The main result of the thesis is a partial differential equation for the fair value of a correlation swap. It will be shown that the expression for the value of a correlation swap obtained by Bossu's model satisfies this partial differential equation.

In this report we study Markov processes on compact and connected Riemannian manifolds. We define a random walk on such manifolds and give a direct proof of the invariance principle. This principle says that under some conditions on the jumping distributions (i.e. the distributions of single steps), the random walk converges to Brownian motion when space is scaled by 1/N, time by N^2 and N tends to infinity (which has been shown with more general methods by Jorgensen and by Blum). To prove this, we show convergence of the generators on the set of smooth functions and we apply the Trotter-Kurtz theorem (as has been done by Blum, in a rather sketchy way and in a slightly different setting). We also show convergence of the corresponding Dirichlet forms. Then we show that the conditions on the jumping distributions are satisfied if they are compactly supported and have mean 0 and a covariance matrix which is invariant under orthogonal transformations.
Next, we define random grids on a Riemannian manifold and we define random walks on them. We show that their Dirichlet forms converge to the Dirichlet form of Brownian motion, using the results above. We also prove a result that is a bit weaker than convergence of the generators in this case.
Finally, these grids allow us to define the Symmetric Exclusion Process (SEP) on a manifold. Using the convergence results above, we follow the line of a proof of Seppäläinen to show that the hydrodynamic limit of the SEP satisfies the heat equation. Some details still need to be filled in, but we believe that this method will allow us to study interacting particle systems and their hydrodynamic limits on Riemannian manifolds.
Before all of this we start with an introduction to Markov processes, their semigroups and generators. In particular we focus on time-reversible (or symmetric) processes and the Dirichlet form with its properties. We also give an introduction to Riemannian manifolds and related notions.

Deriving the location of the maximum of a Brownian Motion with downward quadratic drift. Proving it is welldefined then finding an algorithmic method to simplify expressions of the moments.