In this thesis we study the use of Piecewise Deterministic Markov Processes (PDMPs), such as the Zig-Zag process and the Bouncy Particle Sampler, in rare event probability estimation. We introduce both processes and illustrate how the methods work. To estimate the rare event probabilities we use the splitting method. We also analyze this method and show how the method can be applied by working out a simple example. To make the connection between PDMPs and the splitting method we introduce a setting which suites both well. We consider an example of rare event probabilities outside a d-dimensional sphere for a Gaussian random variable. We compare the results in time complexity, probability estimation, Effective Sample Size and distribution of thresholds between the use of a standard Metropolis-Hastings algorithm and the Bouncy Particle sampler. Furthermore results are obtained to express the eigenvalues of the stochastic process as a function of the boundary on a restricted domain, in particular for the case of the one-dimensional Zig-Zag process with a zero refreshment intensity.

The thesis itself can be found in the repository of TU Delft (https://repository.tudelft.nl/islandora/object/uuid%3A4033e7b9-0e35-4896-95ec-0e5ad25614ed?collection=education). The code used for the simulations can be found in the different jupyter notebook files. In the file 'Computations.ipynb' you can reproduce almost all estimations and simulations that we used in the thesis. In the file 'PDMP.ipynb' there is the code to redo the computations in chapter 6 of the thesis. The simulations can be very time-consuming, so I added four files with rare event probability estimations, which are used in the notebooks as well.