VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Exit problems of Lévy processes with applications in finance

In this thesis we study the pricing of options of American type in a continuous time setting. We begin with a general introduction where we briefly sketch history and different aspects of the option pricing problem. In the first chapter we consider four perpetual options of American type driven by a geometric Brownian motion: the American put and call, the Russian option and the integral option. We derive their values exploiting properties of Brownian motion and Bessel processes. From a practical point of view perpetual options do not seem of much use, since in practice the time of expiration is always finite. However, following an appealing idea of Peter Carr, we build an approximating sequence of perpetual-type options and prove this converges pointwise to the value of the corresponding finite time American option. Next we compute for the mentioned options the first approximation. The second chapter proposes the class of ``phase type Lévy processes'' as a new model for the stock price. This is a class of jump-diffusions which is dense in all Lévy processes and whose positive and negative jumps form compound Poisson processes with jump distributions of phase type. We illustrate its analytical tractability by pricing the perpetual American put and Russian option under this model. In the third chapter we study the same problems but now for the class of Lévy processes without negative jumps. We restrict ourselves to this class, since it contains already a lot of the rich structure of Lévy processes while still being analytically tractable due to many available results exploiting the fact that the jumps of the Lévy process have one sign. A recent study of Carr and Wu offers empirical evidence supporting the case of a model where the risky asset is driven by a spectrally negative Lévy process. For this class of Lévy processes, we review theory on first exit times of finite and semi-infinite intervals. Subsequently, we determine the Laplace transform of the exit time and exit position from an interval containing the origin of the process reflected at its supremum. The proof relies on Itô -excursion theory. The fourth chapter complements the study of the previous chapter. We find the Laplace transform of the first exit time of a finite interval containing the origin of the process reflected at its infimum. Then we turn our attention to these reflected processes killed upon leaving a finite interval containing zero and determine their resolvent measures. Invoking the R-theory of irreducible Markov chains developed by Tuomen and Tweedie, we are able to give a relatively complete description of the ergodic behaviour of their transition probabilities. The obtained results on Lévy processes in chapters 3 and 4 also have applications in the context of the theories of queueing, dams and insurance risk. Finally, the fifth chapter considers the utility-optimisation problem of an agent that operates in a general semimartingale market and seeks to trade so as to maximise his utility from inter-temporal consumption and final wealth. In this setting existence is established following a direct variational approach. Also a characterisation for the optimal consumption and final wealth plan is given

Rational Krylov subspace methods for phi-functions in exponential integrators

Exponential integrators are a class of numerical methods for stiff systems of differential equations which require the computation of products of so-called matrix phi-functions and vectors. In this thesis, we consider the approximation of these matrix functions times some vector by rational and extended Krylov subspace methods. For arbitrary matrices with a field of values in the left complex half-plane, a uniform approximation is obtained that predicts a sublinear convergence

SDEs, Jumps and Estimates

Long Title: Stochastic Ordinary Differential Equations with Jumps: Theory and Estimates. Chapters: Stochastic Integrals - Initial Approach to SDEs - Estimates of SDEs - Other Formulations of SDEs - SDEs with Reflection - PDE Connections