On the Itô-Alekseev-Gröbner Formula for Stochastic Differential Equations
In this thesis we establish the Itô-Alekseev-Gröbner formula which can be regarded as a generalization of the Alekseev-Gröbner lemma and Itô's lemma. Analogous to the well-known deterministic case of the Alekseev-Gröbner lemma, this formula allows us to estimate the global error between the exact solution of a stochastic differential equation (SDE) and a general Itô process in terms of the local characteristics. In particular, our Itô-Alekseev-Gröbner formula can be applied to derive strong approximation rates for implementable approximations of SDEs. To apply the Itô-Alekseev-Gröbner formula, we need to ensure that there exists a version of the solution processes of the SDE which is twice continuously differentiable in the starting point. In the last part of this thesis, we derive conditions on the coefficient functions which ensure existence of solution versions.