Abstract. We propose a mean-reverting model for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non-Gaussian Ornstein-Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. We demonstrate in a simulation example that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. We also discuss the pricing of European call and put options written on electricity forward contracts.

This paper considers a controlled Itô-Lévy process where the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.Mathematics Subject Classification 2000: 93E20, 60H10, 60HXX, 60J75

We specify a general methodological framework for systemic risk measures via multidimensional acceptance sets and aggregation functions. Existing systemic risk measures can usually be interpreted as the minimal amount of cash needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. An important feature of our approach is the possibility of allocating cash according to the future state of the system (scenario-dependent allocation). We also provide conditions that ensure monotonicity, convexity, or quasiconvexity of our systemic risk measures. K E Y W O R D Sacceptance set, aggregation, systemic risk, risk measures INTRODUCTIONThe financial crisis has dramatically demonstrated that traditional risk management strategies of financial systems, which predominantly focus on the solvency of individual institutions as if they were in isolation, insufficiently capture the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. A large part of the current literature on systemic financial risk is concerned with the modeling structure of financial networks, the analysis of the contagion, and the spread of a potential exogenous (or even endogenous) shock into the system. For a given financial (possibly random) network and a given random shock, one then Mathematical Finance. 2019;29:329-367.wileyonlinelibrary.com/journal/mafi

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 our previous paper, "A Unified Approach to Systemic Risk Measures via Acceptance Set" (Mathematical Finance, 2018 ), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure.In addition, the optimizer in the dual formulation provides a risk allocation which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.