Euler schemes and large deviations for stochastic Volterra equations

Abstract

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.