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A. Grass:
"Uniqueness Properties of Barrier Type Skorokhod Embeddings and Perkins Embedding with General Starting Law";
Supervisor: M. Beiglböck; 105.7 MSTOCH, Institut für Stochastik und Wirtschaftsmathematik, 2017.

Abstract. Since its formulation in 1961 the Skorokhod embedding problem - that is to represent a given probability measure as a standard Brownian motion B stopped at a specific stopping time - has gained more and more fame until now even being considered a classical problem in probability theory. With new solution emerging every couple of years, the theory recently enjoyed (partial)
unification in [BCH17].
A major inspiration for a lot of publications regarding the Skorokhod embedding problem was the idea of Root to identify solutions as hitting times of certain subsets of R+ ⇥ R. These solutions do not only have a nice geometric interpretation but also come with a uniqueness property.
[BCH17] gives the tools to identify a lot of known solutions as Root solutions in the sense that these solutions can be represented as hitting times of a process (A, B), where the choice of the process A will depend on the additional properties of the specific solutions.
The first objective of this thesis is to show rigorously that the uniqueness property of the original Root solution carries over to this more general setup - even if considering a generalized Skorokhod embedding problem where we embed in processes different from a Brownian Motion.
Another development in the theory around the Skorokhod embedding problem (often with regard to application in financial mathematics) is to embed into a Brownian motion started according to a nontrivial initial distribution.
In the second part of this thesis we will consider a specific solution to the Skorokhod embedding problem proposed by Perkins in [Per86].
For a simplified case, that is, where the initial and the terminal solution do not share any mass, we will by using methods given in [BCH17] to find a solution to the Perkins embedding with random starting which is given as a hitting time in sense of a generalized Root solution. We will also establish a uniqueness result in this context.