Disputas: David Ruiz Baños
M. Sc. David Ruiz Baños ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:
Regularity of Stochastic Flows of Stochastic Differential Equations with Singular Coefficients and Applications to Finance
David Ruiz Baños
Tid og sted for prøveforelesning
- Professor José Manuel Corcuera Valverde, Universitat de Barcelona
- Associate Professor Arne Løkka, London School of Economics
- Professor Tom Louis Lindstrøm, Universitetet i Oslo
Leder av disputas
Instituttleder Arne Huseby, Matematisk institutt, Universitet i Oslo
- Professor Frank Norbert Proske, Matematisk institutt, Universitet i Oslo
- Professor Giulia Di Nunno, Matematisk institutt, Universitet i Oslo
- Professor Bernt Øksendal, Matematisk institutt, Universitet i Oslo
The study of nature and our environment is a large part of science. Life science, social or physical sciences constitute important branches of science. The scientific method can be regarded as a collection of techniques for investigating phenomena in the aforementioned areas of research, acquiring knowledge, correcting or updating with new knowledge and drawing conclusions or predictions. Mathematics establishes the grounds of logic reasoning to develop theoretical models and rules of calculus that serve as framework and utensils to explain such phenomena. An example of this is mathematical models such as differential equations, which, in short, can be viewed as theoretical laws describing some phenomenon of interest. For the reason that nature is complex, sometimes, one needs to consider random inputs in the model that may interfere in, say measurements or possible outcomes or results. In the recent decades the branch stochastic analysis has been a broadly investigated topic as a possible answer for this, especially within economical sciences. In this thesis we focus in this particular branch of mathematics: stochastic calculus and stochastic differential equations, which in a summary, comprise the rules of calculus one needs to deal with theoretical models involving “randomness” (stochastic differential equations). Because real life problems are extremely complex and difficult to model, in this thesis, we study all these objects in what we call an “irregular setting”, that is to say, when the parts of the model behave in a “rough way” with sudden changes and jumps. Then the natural question whether one can solve such equations in this irregular setting arises and what properties of the solutions of these equations we can expect. In this thesis we answer these questions. We also find an application to mathematical finance in the so-called sensitivity analysis of financial options.
In this thesis we use a new method to construct solutions of stochastic differential equations when the coefficients involved are singular. We study their regularity in several aspects and at several levels. We study the regularity of flows of the solution and thus extend some existing results on this topic. We also provide with a numerical method to compute sentivity with respect to initial conditions based on the so-called Malliavin calculus and local-time integration under presence or singular coefficients. Finally, we study the problem of changing the source of noise in the stochastic equation by fractional Brownian motion which is neither a Markov process nor a weak semimartingale and construct for the first time solutions of such equations when the coefficient is singular and dimensions are high.
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