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Parametrix Method and its Applications in Probability Theory

Kozhina, Anna

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Abstract

The present thesis investigates applications and developments of the parametrix technique. The parametrix technique comes from the theory of ODEs. Now it reformulates as a continuity technique that provides a formal representation for the density of the SDE's solutions in terms of infinite series involving the density of another, simpler, Markov process. Although the method itself has been known for many years there are still many open problems. The project is divided into three parts. Firstly, we are going to introduce the parametrix method for diffusions and Markov chains in general settings. After presenting main concepts and objects we emphasize that the technique can be fruitfully used also in case of non-smooth coeffcients. Secondly, we study the sensitivity of densities of non-degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coeffcients. Natural applications of these results appear, for example, in models with misspecifed coeffcients or for the investigation of the weak error of the Euler scheme with irregular coeffcients. The stability controls have been derived under Hölder continuity assumptions on coeffcients regularity only. Continuing the research, V. Konakov and S. Menozzi applied the results mentioned above to study the weak error of the Euler scheme approximations in their paper[V. Konakov, S. Menozzi, 2017, Weak Error for the Euler Scheme Approximation of Diffusions with non-smooth coeffcients]. Motivated by these extensions, we continue with the most challenging and diffcult part - the weak and global error controls for the case of rough coefficients to Kolmogorov's degenerate SDEs in the last part of the thesis. Such equations were first introduced in 1933 by Kolmogorov. Adapting the techniques, introduced in the paper written by V. Konakov, S Menozzi and S. Molchanov in 2010 (where authors considered Lipschitz coefficients), it is now possible to investigate the Holder settings for degenerate Kolmogorov diffusions. To specify the notation of the weak and global error in our framework, we also introduce the specific version of the Euler scheme for the degenerate Kolmogorov equation, which can be also seen as an Ito process. The sensitivity analysis which we need to prove controls for the weak end global errors naturally extends from the non-degenerate case to the degenerate framework. However, some structural difficulties appear due to the different time scales for the first and the second space variable of the transition density. These aspects can be also found in the published article [A. Kozhina, 2016, Stability of densities for perturbed degenerate diffusions]. One of the key results in the last part provides the weak error controls for degenerate diffusions with non-smooth coefficients. To derive that we have proved the heat kernel derivatives bounds with respect to a non-degenerate first component of the transition density. Up to the best of our knowledge, these are the first pointwise bounds obtained on the derivatives w.r.t. the non-degenerate variables under the sole Hölder continuity assumption on the coefficients. They extend the well-known controls derived by Il'in et al. in 1962 to Kolmogorov diffusions. Investigating the quantitative behaviour of the derivatives w.r.t. the degenerate variable under minimal smoothness assumptions remains a very interesting and open problem. Finally, we studied the controls for the direct difference of transition densities of the diffusion and the Markov chain. Unfortunately, when handling directly the difference of the densities we cannot avoid to control sensitivities of the kernels w.r.t. to the degenerate variable. Such sensitivities lead to higher time singularities and make the unbounded transport term appear. The higher time-singularities yield the stated restriction on the Hölder exponent in our assumptions on the coefficients.

Item Type: Dissertation
Supervisor: Mammen, Prof. Dr. Enno
Place of Publication: Heidelberg
Date of thesis defense: 16 July 2018
Date Deposited: 15 Aug 2018 10:15
Date: 2018
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
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