1. INTRODUCTION 5

for the increment-tamed Euler-Maruyama scheme (1.5). Moreover, Corollary 2.27

in Subsection 2.3.1 yields uniform moment bounds for the fully drift-implicit Euler

scheme and Lemma 2.28 in Subsection 2.3.2 establishes uniform moment bounds

for partially drift-implicit approximation schemes. These results on uniform mo-

ment bounds are applications of a general theory which we develop in Section 2.1.

In this theory (see Propositions 2.1 and 2.7 and Corollaries 2.2, 2.3 and 2.6) we

assume a Lyapunov-type inequality to be satisfied by the approximation processes

on large subevents of the probability space, i.e., on complements of rare events; see

inequality (2.11) in Corollary 2.2. One of our main results (Theorem 2.13 in Sub-

section 2.2.1) establishes this Lyapunov-type condition for the Euler-Maruyama

approximations (1.4). More precisely, whereas the Euler-Maruyama approxima-

tions often do not satisfy a Lyapunov-type inequality on events of probability one

in the case of superlinearly growing coeﬃcients according to Corollary 2.17 in Sub-

section 2.2.1, the Euler-Maruyama approximations do satisfy the Lyapunov-type

inequality (2.11) on large subevents of the probability space according to Theo-

rem 2.13 in Subsection 2.2.1. This integrability result on the Euler-Maruyama

approximation processes can then be transfered to a large class of other one-step

approximation processes. To be more precise, Lemma 2.18 in Subsection 2.2.2

proves that if two general one-step approximation schemes are close to each other

in the sense of (2.102) (see Lemma 2.18 for the details) and if one approximation

scheme satisfies the Lyapunov-type inequality (2.11) on large subevents, then the

other approximation scheme satisfies the Lyapunov-type inequality (2.11) on large

subevents of the probability space as well. After having established the Lyapunov-

type inequality (2.11) on such complements of rare events, the general rare event

based theory in Section 2.1 can be applied to derive moment bounds and further

integrability properties of the approximation processes. In Chapter 3, we then pro-

ceed to study convergence in probability (see Section 3.3), strong convergence (see

Section 3.4) and weak convergence (see Section 3.5) of approximation processes for

SDEs. Definition 3.1 in Section 3.2 specifies a local consistency condition on ap-

proximation schemes which is, according to Theorem 3.3 in Section 3.3, suﬃcient for

convergence in probability of the approximation processes to the exact solution of

the SDE (1.1). This convergence in probability and the uniform moment bounds in

Corollary 2.21 then result in the strong convergence (1.7) of the increment-tamed

Euler-Maruyama approximations (1.5); see Theorem 3.15 in Subsection 3.4.3 for

the details. Moreover, we obtain results for approximating moments and more gen-

eral statistical quantities of solutions of SDEs of the form (1.1) in Section 3.5. In

particular, Corollary 3.23 in Subsection 3.5.2 establishes convergence of the Monte

Carlo Euler approximations for SDEs of the form (1.1).

1.1. Notation

Throughout this article, the following notation is used. For a set Ω, a measur-

able space (E, E) and a mapping Y : Ω → E we denote by σΩ(Y ) := {Y

−1(A)

⊂

Ω: A ∈ E} the smallest sigma algebra with respect to which Y : Ω → E is measur-

able. Furthermore, for a topological space (E, E) we denote by B(E) := σE(E) the

Borel sigma-algebra of (E, E). Moreover, for a natural number d ∈ N and two sets