CHAPTER 1

Introduction

This article investigates integrability and convergence properties of numerical

approximation processes for stochastic differential equations (SDEs). In order to

illustrate one of our main results, the following general setting is considered in this

introductory chapter. Let T ∈ (0, ∞), d, m ∈ N := {1, 2,...}, let (Ω, F, P) be

a probability space with a normal filtration (Ft)t∈[0,T

]

, let W : [0,T ] × Ω → Rm

be a standard (Ft)t∈[0,T

]

-Brownian motion, let D ⊂

Rd

be an open set, let μ =

(μ1,...,μd): D →

Rd

and σ = (σi,j)i∈{1,2,...,d},j∈{1,2,...,m} : D →

Rd×m

be locally

Lipschitz continuous functions and let X : [0,T ] × Ω → D be an (Ft)t∈[0,T

]

-adapted

stochastic process with continuous sample paths satisfying the SDE

(1.1) Xt = X0 +

t

0

μ(Xs) ds +

t

0

σ(Xs) dWs

P-almost surely for all t ∈ [0,T ]. Here μ is the infinitesimal mean and σ ·

σ∗

is

the infinitesimal covariance matrix of the solution process X of the SDE (1.1). To

guarantee finiteness of some moments of the SDE (1.1), we assume existence of a

Lyapunov-type function. More precisely, let q ∈ (0, ∞), κ ∈ R be real numbers and

let V : D → [1, ∞) be a twice continuously differentiable function with E[V (X0)]

∞ and with V (x) ≥ x

q

and

(1.2)

d

i=1

∂V

∂xi

(x) · μi(x) +

1

2

d

i,j=1

m

k=1

∂2V

∂xi∂xj

(x) · σi,k(x) · σj,k(x) ≤ κ · V (x)

for all x ∈ D. These assumptions ensure

(1.3) E V (Xt) ≤

eκt

· E V (X0)

for all t ∈ [0,T ] and, therefore, finiteness of the q-th absolute moments of the

solution process Xt, t ∈ [0,T ], of the SDE (1.1), i.e., supt∈[0,T

]

E Xt

q

∞.

Note that in this setting both the drift coeﬃcient μ and the diffusion coeﬃcient σ

of the SDE (1.1) may grow superlinearly and are, in particular, not assumed to be

globally Lipschitz continuous. Our main goal in this introduction is to construct and

to analyze numerical approximation processes that converge strongly to the exact

solution of the SDE (1.1). The standard literature in computational stochastics

(see, for instance, Kloeden & Platen [47] and Milstein [61]) concentrates on SDEs

with globally Lipschitz continuous coeﬃcients and can therefore not be applied

here. Strong numerical approximations of the SDE (1.1) are of particular interest

for the computation of statistical quantities of the solution process of the SDE (1.1)

through computationally eﬃcient multilevel Monte Carlo methods (see Giles [20],

Heinrich [29] and, e.g., in Creutzig et al. [13], Hickernell et al. [32], Barth, Lang

& Schwab [5] and in the references therein for further recent results on multilevel

Monte Carlo methods).

1