Title: Euler schemes and large deviations for stochastic Volterra equations
Other Titles: Euler-Verfahren und große Abweichungen für stochastische Volterragleichungen
Language: English
Authors: Münz, Philip 
Qualification level: Diploma
Advisor: Gerhold, Stefan 
Issue Date: 2020
Number of Pages: 32
Qualification level: Diploma
The main goal of this thesis is to show the Large Deviation Principle (LDP, see definition 4.2) for a family $\{X^\varepsilon, \varepsilon > 0\}$ where each $X^\varepsilon$ is solving a stochastic Volterra integral equation of the form\begin{equation*}X^\varepsilon_t = X^\varepsilon_0 + \int_0^t b(t,s,X^\varepsilon_s) \ dt + \sqrt{\varepsilon} \int_0^t \sigma(t,s,X^\varepsilon_s) \ dW\end{equation*}on the same probability space where $W$ is a Standard Brownian Motion. Chapter 2 contains the notations which will be used and in section 2.3 the assumptions under which the statements of this thesis hold are listed. The proof of the LDP will be done in chapter 5 by showing the Laplace Principle. The equivalence of these two principles and the conditions under which this equivalence holds true is stated in chapter 4 (see also [DE11]). In chapter \ref{Konvergenz von Xn} an Euler scheme for this type of integral equation is presented.\newline\hspace*{5mm}A large part of this thesis is dedicated to giving more detailed proofs of the statements from [Zha08], some of which are shown under stronger conditions than in the corresponding paper, since the proofs in [Zha08] for the weaker ones were not completely clear to me.
Keywords: Stochastische Volterra-Integralgleichung; Euler-Verfahren; Prinzip der Großen Abweichungen; Laplace-Prinzip
Stochastic Volterra Integral Equation; Euler scheme; Large Deviation Principle; Laplace Principle
URI: https://doi.org/10.34726/hss.2020.80522
DOI: 10.34726/hss.2020.80522
Library ID: AC16099032
Organisation: E105 - Institut für Stochastik und Wirtschaftsmathematik 
Publication Type: Thesis
Appears in Collections:Thesis

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