This thesis has been funded by Center of Mathematics for Applications (CMA) and the Mathematical Institute both at the University of Oslo. During four years I have had the pleasure of being a part of the Stochastic Analysis group which has given me the opportunity to meet and work with a number of inspiring people. First and foremost, I would like to thank my primary supervisor, Frank Proske. It is difficult to explain how important his guidance has been for me. For me, there could not be a better supervisor. Our hundreds of talks over hundreds of cups of coffee has always left me motivated and inspired. I would also like to thank all the people at the CMA for creating such an inspiring environment. In particular I want to thank David Baños, Giulia di Nunno, Sven Haadem, Erlend Storrøsten and Bernt Øksendal for valuable discussions as well as my coauthors Franco Flandoli, Olivier Menokeu Pamen, Thilo Meyer-Brandis, Salah Mohammed and Tusheng Zhang. Also the administrative staff, in particular Biljana Dragisic, Robin Jacobsen and Elisabeth Seland deserves a thank you for making bureaucracy a less painful experience. Between January and July 2013 I spent my time at Humboldt University of Berlin. I would like to thank Peter Imkeller and his research group for including me in Berlin. In particular, I am grateful to Nicolas Perkowski for interesting discussions. Finally, I would like to thank the most important person in my life, Ellen, for always being loving and supportive.
In this paper, we consider a numerical approximation of the stochastic differential equation (SDE)where the drift coefficient b :Hölder continuous in both time and space variables and the noise L = (Lt) 0≤t≤T is a d-dimensional Lévy process. We provide the rate of convergence for the Euler-Maruyama approximation when L is a Wiener process or a truncated symmetric α-stable process with α ∈ (1, 2). Our technique is based on the regularity of the solution to the associated Kolmogorov equation.
In this paper, we derive a general stochastic maximum principle for a risk-sensitive type optimal control problem of Markov regime-switching jump-diffusion model. The results are obtained via a logarithmic transformation and the relationship between adjoint variables and the value function. We apply the results to study both a linear-quadratic optimal control problem and a risk-sensitive benchmarked asset management problem for Markov regime-switching models. In the latter case, the optimal control is of feedback form and is given in terms of solutions to a Markov regime-switching Riccatti equation and an ordinary Markov regime-switching differential equation.
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