Introduction

Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 67)

Abstract

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. 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. 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.

Keywords

Brownian Motion Central Limit Theorem Quadratic Variation Functional Versus Polynomial Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Paris VI – Pierre et Marie CurieParisFrance
  2. 2.Department of StatisticsColumbia UniversityNew YorkUSA

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