2012 | OriginalPaper | Chapter
Introduction
Authors : Jean Jacod, Philip Protter
Published in: Discretization of Processes
Publisher: Springer Berlin Heidelberg
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This chapter is an introduction for the methods and content of the book. After a brief description of the book’s contents, we give results in a simple setting: the underlying process
X
is a one-dimensional Lévy process which is either continuous, or has finitely many jumps in finite intervals. The process is discretized along a regular grid of mesh
Δ
n
which eventually goes to 0, and we introduce two kinds of functionals of interest for this setting:
1.
The “unnormalized functional”
$V^{n}(f,X)_{t}=\sum_{i=1}^{[t/\varDelta _{n}]}f(X_{i\varDelta _{n}}-X_{(i-1)\varDelta _{n}})$
for any given test function
f
, and where [
t
/
Δ
n
] denotes the integer part of
t
/
Δ
n
.
2.
The “normalized functional”
$V'^{n}(f,X)_{t}=\varDelta _{n}\sum_{i=1}^{[t/\varDelta _{n}]}f((X_{i\varDelta _{n}}-X_{(i-1)\varDelta _{n}})/\sqrt{\varDelta _{n}}\,)$
, where the (inside) normalized factor
$\sqrt{\varDelta _{n}}$
is chosen so that when
X
is a Brownian motion the argument of
f
in each summand is a standard normal variable.
Then, with
X
as described above, we explain the sort of limiting behavior one may expect for these functionals: the convergence in probability towards a suitable limit, and the associated Central Limit Theorem. The simple setting allows one to give a heuristic explanation of the results, and of the conditions on the test function
f
which are necessary to obtain these results.