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
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