Excessive functions, Appell polynomials and optimal stopping

The main topic of the thesis is optimal stopping. This is treated in two research articles. In the first article we introduce a new approach to optimal stopping of general strong Markov processes. The approach is based on the representation of excessive functions as expected suprema. We present a variety of examples, in particular, the Novikov-Shiryaev problem for Lévy processes. In the second article on optimal stopping we focus on differentiability of excessive functions of diffusions and apply these results to study the validity of the principle of smooth fit. As an example we discuss optimal stopping of sticky Brownian motion. The third research article offers a survey like discussion on Appell polynomials. The crucial role of Appell polynomials in optimal stopping of Lévy processes was noticed by Novikov and Shiryaev. They described the optimal rule in a large class of problems via these polynomials. We exploit the probabilistic approach to Appell polynomials and show that many classical results are obtained with ease in this framework. In the fourth article we derive a new relationship between the generalized Bernoulli polynomials and the generalized Euler polynomials.

Spectroscopic Studies of III-V Semiconductor Materials for Improved Devices

Defects in semiconductor crystals and at their interfaces usually impair the properties and the performance of devices. These defects include, for example, vacancies (i.e., missing crystal atoms), interstitials (i.e., extra atoms between the host crystal sites), and impurities such as oxygen atoms. The defects can decrease (i) the rate of the radiative electron transition from the conduction band to the valence band, (ii) the amount of charge carriers, and (iii) the mobility of the electrons in the conduction band. It is a common situation that the presence of crystal defects can be readily concluded as a decrease in the luminescence intensity or in the current flow for example. However, the identification of the harmful defects is not straightforward at all because it is challenging to characterize local defects with atomic resolution and identification. Such atomic-scale knowledge is however essential to find methods for reducing the amount of defects in energy-efficient semiconductor devices. The defects formed in thin interface layers of semiconductors are particularly difficult to characterize due to their buried and amorphous structures. Characterization methods which are sensitive to defects often require well-defined samples with long range order. Photoelectron spectroscopy (PES) combined with photoluminescence (PL) or electrical measurements is a potential approach to elucidate the structure and defects of the interface. It is essential to combine the PES with complementary measurements of similar samples to relate the PES changes to changes in the interface defect density. Understanding of the nature of defects related to III-V materials is relevant to developing for example field-effect transistors which include a III-V channel, but research is still far from complete. In this thesis, PES measurements are utilized in studies of various III-V compound semiconductor materials. PES is combined with photoluminescence measurements to study the SiO2/GaAs, SiNx/GaAs and BaO/GaAs interfaces. Also the formation of novel materials InN and photoluminescent GaAs nanoparticles are studied. Finally, the formation of Ga interstitial defects in GaAsN is elucidated by combining calculational results with PES measurements.