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2012 | Book

Introduction Read chapter Read first chapter

Discretization of Processes

Authors: Jean Jacod, Philip Protter

Publisher: Springer Berlin Heidelberg

Book Series : Stochastic Modelling and Applied Probability

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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About this book

In applications, and especially in mathematical finance, random time-dependent events are often modeled as stochastic processes. Assumptions are made about the structure of such processes, and serious researchers will want to justify those assumptions through the use of data. As statisticians are wont to say, “In God we trust; all others must bring data.”

This book establishes the theory of how to go about estimating not just scalar parameters about a proposed model, but also the underlying structure of the model itself. Classic statistical tools are used: the law of large numbers, and the central limit theorem. Researchers have recently developed creative and original methods to use these tools in sophisticated (but highly technical) ways to reveal new details about the underlying structure. For the first time in book form, the authors present these latest techniques, based on research from the last 10 years. They include new findings.


This book will be of special interest to researchers, combining the theory of mathematical finance with its investigation using market data, and it will also prove to be useful in a broad range of applications, such as to mathematical biology, chemical engineering, and physics.

Table of Contents

Frontmatter

Introduction and Preliminary Material

Frontmatter
Chapter 1. Introduction
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.
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.
Jean Jacod, Philip Protter
Chapter 2. Some Prerequisites
Abstract
This second preliminary chapter is devoted to recalling some basic properties of semimartingales, and it is about the convergence of processes. Both of these topics are prerequisites for the rest of the book.
In Sect. 2.1 the main properties of semimartingales are recalled, with a special emphasis on a description of the so-called Itô semimartingales. We also recall basic features of the characteristics of a semimartingale. Most of the results established in this book hold only for Itô semimartingales. Almost all the properties of semimartingales we review here can be found in various books already, so proofs are omitted. A few results are new in book form, and these are proved in the Appendix: this mainly concerns a number of estimates on the increments of a semimartingale, or an Itô semimartingale. These results are scattered in various papers but are essential to our aims, so it is prudent to collect them here.
Section 2.2 is devoted to recalling facts about the convergence of processes, starting with a quick reminder about the Skorokhod topology. Here, special emphasis is put on stable convergence in law, which is central for almost all statistical applications of the results of this book. Again, the proofs of most results, which can be found in various books, is omitted, whereas the proofs of the few results which are new in book form is given in the Appendix.
Jean Jacod, Philip Protter

The Basic Results

Frontmatter
Chapter 3. Laws of Large Numbers: The Basic Results
Abstract
In this chapter the Laws of Large Numbers for the two types of functionals V n (f,X) and V′ n (X,f) are provided. By this, we mean their convergence in probability, usually for the Skorokhod topology. An important feature should be mentioned: unlike the case of the “usual” Law of Large Numbers, the limit is typically not deterministic, but random.
As seen in the text, the Law of Large Numbers for the unnormalized functionals V n (f,X) holds for all semimartingales, and for arbitrary discretization schemes, under appropriate conditions on the test function f, of course. This can be viewed as an extension of the well known property that the approximate quadratic variation of a semimartingale X, that is the sum of the squared increments of X taken along an increasing sequence of stopping times, converges in probability to the “true” quadratic variation [X,X] of X when the mesh of the discretization scheme goes to 0.
In contrast, the Law of Large Numbers for the normalized functionals V′ n (f,X), in which the argument of the test function f is taken to be the increment of X on each discretization interval, divided by the square-root of its length, holds only for Itô semimartingales and for regular discretization schemes (although an extension is presented in Chap. 14 later, for some special irregular discretization grids). An interesting feature is that the limiting process depends only on the Brownian part of X, and more specifically on the volatility process.
The chapter starts with two preliminary sections, about “general” discretization schemes, and about semimartingales that have p-summable jumps, including an extension of Itô’s formula to functions that are not necessarily C 2. It also ends with a quick description of two applications, which are studied more thoroughly in the next chapters: the estimation of the “integrated volatility” for a continuous Itô semimartingale, and the detection of jumps for an Itô semimartingale.
Jean Jacod, Philip Protter
Chapter 4. Central Limit Theorems: Technical Tools
Abstract
The aim of this chapter is to establish the theoretical basis for the Central Limit Theorems associated with the Laws of Large Numbers of the previous chapter.
The reason for presenting this material in a separate chapter is that Central Limit Theorems have rather long proofs, but for the functionals previously considered, as well as for more general functionals to be seen in the forthcoming chapters, the proofs are always based on the same ideas and techniques; we thus thought more suitable to present these general ideas and technical tools in a specific chapter.
These tools will be in constant use later, hence the results of this chapter are considerably more general than strictly needed for the functionals V n (f,X) and V′ n (f,X) presented before. However, although this greater generality complicates the notation, it creates only very few supplementary technicalities.
Section 4.1 is devoted to a description of the limiting processes occurring in the various Central Limit Theorems, and which are processes with “conditionally independent increments”. Sections 4.2 and 4.3 provide general criteria for stable convergence in law, when the limit is a continuous process (Sect. 4.3) or a possibly discontinuous one (Sect. 4.3). These results are given for sums of triangular arrays, and the application to discretized Itô semimartingales is presented in Sect. 4.4. This last section also contains a description of the so-called “localization procedure” which allows us to reduce the problem to the case where the semimartingale has nice properties, namely the boundedness of its characteristics.
Jean Jacod, Philip Protter
Chapter 5. Central Limit Theorems: The Basic Results
Abstract
This chapter is in a sense the central part of the book: it contains the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 3: the one for the unnormalized functionals V n (f,X) is in Sect. 5.1, whereas Sects. 5.2 and 5.3 provide the ones for the normalized functionals V′ n (f,X). In both cases, X needs to be an Itô semimartingale, and only regular discretization schemes are considered.
Section 5.4 contains the Central Limit Theorem for the discrete approximations of the quadratic variation, again along regular schemes. Surprisingly (or perhaps, not so surprisingly), and although this amounts to studying the functionals V n (f,X) or V′ n (f,X) for the test function f(x)=x 2 (the two kinds of functionals are then identical), the assumptions on X needed for this case are significantly weaker than they are for more general test functions.
We also provide in Sect. 5.5 a “joint Central Limit Theorem” for the pair (V n (f,X),V′ n (g,X)), with f and g two (possibly multi-dimensional) test functions.
Finally, Sect. 5.6 is devoted to some applications: first we pursue the estimation of the “integrated volatility” for a continuous Itô semimartingale, and the detection of jumps for an Itô semimartingale, started in Chap. 3, and for example give confidence bounds for the integrated volatility, or tests for deciding whether the process X is continuous or discontinuous. Second, we show how the Central Limit Theorem for the quadratic variation can be used for studying Euler schemes for stochastic differential equations which are driven by a general Itô semimartingale X; in particular, we give a Central Limit Theorem for the error incurred in this method, that is for the difference Y n −Y where Y is the solution of the equation and Y n is its Euler approximation with (for example) step size \(\frac{1}{n}\).
Jean Jacod, Philip Protter
Chapter 6. Integrated Discretization Error
Abstract
In this chapter, which slightly deviates from the general topic of this book, we study another type of functionals. Namely, if \(X^{n}_{t}=X_{[t/ \varDelta _{n}]}\) denotes the process obtained by discretization of the Itô semimartingale X along a regular grid with stepsize Δ n , we study the integrated error: this can be \(\int_{0}^{t}(f(X^{n}_{s})-f(X_{s}))\,ds\) or, in the L p sense, \(\int_{0}^{t}|f(X^{n}_{s})-f(X_{s})|^{p}\,ds\).
In both cases, and if f is C 2, these functionals, suitably normalized, converge to a non-trivial limiting process. In the first case, the proper normalization is 1/Δ n , exactly as if X were a non-random function with bounded derivative. In the second case, one would expect the normalizing factor to be \(1/ \varDelta _{n}^{p/2}\), at least when p≥2: this is what happens when X is continuous, but otherwise the normalizing factor is 1/Δ n , regardless of p≥2.
Jean Jacod, Philip Protter

More Laws of Large Numbers

Frontmatter
Chapter 7. First Extension: Random Weights
Abstract
In this chapter a Law of Large Numbers is proved for an extension of the functionals V n (f,X) and V′ n (f,X). For the first one, the summands \(f(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})\) are replaced by \(F(\omega ,(i-1) \varDelta _{n},X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})\) for a function F on Ω×ℝ+×ℝ d , where d is the dimension of X, and likewise for the functional V′ n (f,X). The results are perhaps obvious generalizations of those of Chap. 3, the main difficulty being to establish the assumptions on F ensuring the convergence.
The motivation for this is to solve parametric statistical problems for discretely observed processes: in the last section one considers the solution of a (continuous) stochastic differential equation whose diffusion coefficient depends smoothly on a parameter θ. Then, on the basis of discrete observation along a regular grid, one shows how the previous Laws of Large Numbers can be put to use for constructing estimators of θ which are consistent, as the discretization mesh goes to 0.
Jean Jacod, Philip Protter
Chapter 8. Second Extension: Functions of Several Increments
Abstract
The aim of this chapter is to extend the Laws of Large Numbers to functionals in which each summand depends on several successive increments of the underlying process X. This covers two different situations:
1.
The test function f is replaced by a function F on (ℝ d ) k , where d is the dimension of X and k≥2 is an integer. Then the ith summand in the unnormalized functional is \(F(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}},\dots,X_{(i+k-1)\varDelta _{n}},-X_{(i-k-2) \varDelta _{n}})\), and the same for the normalized functional, upon dividing each increment by \(\sqrt {\varDelta _{n}}\).
 
2.
Each summand is a function of k n successive increments (after dividing by \(\sqrt {\varDelta _{n}}\) for the normalized functionals), where k n is a sequence of integers increasing to ∞, but such that k n Δ n →0. This poses a formulation problem which is presented in Sect. 8.1, because then the test function must depend on n because its argument is k n successive increments, and a form of “compatibility” for different values of n has to be assumed.
 
In Sects. 8.2 and 8.3 the Laws of Large Numbers for the unnormalized functionals are presented, for a fixed number k or an increasing number k n of increments, respectively: the methods and results are deeply different in the two cases. In contrast, the results for the normalized functionals, given in Sect. 8.4, are basically the same for a fixed number k or an increasing number k n of increments.
Of particular interest is the case of a fixed number k of increments, when the test function has a product form, for example when d=1 it could be \(f(x_{1},\dots,x_{k})=|x_{1}|^{p_{1}}\cdots|x_{k}|^{p_{k}}\) for positive reals p j . The associated functionals are then called multipower variations and have been extensively used for estimating the integrated volatility when the process X has jumps. This application is presented from the consistency viewpoint here. The Central Limit Theorem is studied later and is given in Sect. 8.5.
Jean Jacod, Philip Protter
Chapter 9. Third Extension: Truncated Functionals
Abstract
Here one studies “truncated functionals”, in which a summand in the definition of the functional is discarded when the corresponding increment is smaller (upward truncation) or bigger (downward truncation) in absolute value than some level v n >0. This level v n depends on the mesh Δ n and typically goes to 0 as Δ n →0. This allows one to disentangle the “jump part” and the “Brownian part” of the Itô semimartingale: when interested by jumps, one considers the upward truncation, and one uses downward truncation when one wants to retrieve the Brownian part.
In Sect. 9.1 the upward truncated unnormalized functionals are studied. The Law of Large Numbers for downward truncated normalized functionals, including the case of dependence on several increments, is given in Sect. 9.2.
Sections 9.3, 9.4 and 9.5 are concerned with a “local approximation” of the volatility, using downward truncated normalized functionals: assuming a suitable regularity of the volatility process σ t , the aim is to estimate σ t (or rather its “square” \(\sigma _{t}\sigma _{t}^{*}\)). Statistical applications are given in Sect. 9.6.
Jean Jacod, Philip Protter

Extensions of the Central Limit Theorems

Frontmatter
Chapter 10. The Central Limit Theorem for Random Weights
Abstract
This chapter presents the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 7: the summands in the unnormalized functionals are now
$$F\bigl(\omega ,(i-1) \varDelta _n,X_{i \varDelta _n}-X_{(i-1) \varDelta _n}\bigr)$$
for a function F on Ω×ℝ+×ℝ d , where d is the dimension of X, and it is the same for the normalized functional upon dividing the increment by \(\sqrt {\varDelta _{n}}\).
Sections 10.1 and 10.2 are devoted to unnormalized functionals, in two situations: first we treat the case for a “general” test function F, satisfying rather strong regularity assumptions as a function of time; and second, we treat the case for F of the form F(ω,t,x)=G(X t (ω),x), where G is a (smooth enough) function on (ℝ d )2. The same task is performed for normalized functionals in Sect. 10.3, again in the two cases mentioned before.
Finally, in Sect. 10.4 we present an application to the estimation of a parameter θ for the solution of a (continuous) stochastic differential equation whose diffusion coefficient depends smoothly on θ, and which is observed at the discrete times iΔ n over a finite time interval [0,T]. In particular, we show how to construct estimators which are asymptotically (mixed) normal with the optimal rate of convergence \(\sqrt {\varDelta _{n}}\).
Jean Jacod, Philip Protter
Chapter 11. The Central Limit Theorem for Functions of a Finite Number of Increments
Abstract
In this chapter we give the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 8, when the number k of increments in the test function is fixed.
For unnormalized functionals, studied in Sect. 11.1, this is a rather straightforward extension of the Central Limit Theorems given in Chap. 5.
In Sect. 11.2, normalized functionals are considered. In this case, the situation is much more complicated than in Chap. 5, because two successive summands in the definition of the functional involve k−1 “common” increments of the process X. Joint Central Limit Theorems for the two types of functionals are presented in Sect. 11.3.
Finally, in Sect. 11.4 we present some statistical applications, both for the estimation of the volatility and for the detection of jumps of the process X. Using functions of several increments (and in particular multipower variations) allows one to estimate the integrated volatility even when the process X has jumps.
Jean Jacod, Philip Protter
Chapter 12. The Central Limit Theorem for Functions of an Increasing Number of Increments
Abstract
Here we study the same problem as in the previous chapter, except that now the functionals depend on an increasing number k n of increments, with kj n →∞ and k n Δ n →0.
In this setting, the Central Limit Theorems are considerably more difficult to prove, and the rate of convergence becomes \(\sqrt{k_{n} \varDelta _{n}}\) instead of \(\sqrt {\varDelta _{n}}\). Unnormalized and normalized functionals are studied in Sects. 12.1 and 12.2, respectively.
No specific application is given in this chapter, but it is a necessary step for studying semimartingales contaminated by an observation noise, and we treat this in Chap. 16.
Jean Jacod, Philip Protter
Chapter 13. The Central Limit Theorem for Truncated Functionals
Abstract
In this chapter we prove the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 9, about truncated functionals.
Section 13.1 is devoted to unnormalized functionals, truncated upward, whereas Sect. 13.2 is about normalized functionals, truncated downward, and we include functionals depending on k successive increments. In both cases, the proofs are straightforward extensions of those in Chaps. 5 and 11.
In Sect. 13.3 one studies the rate of convergence of the “local approximations” of the volatility, introduced in Chap. 9. This part necessitates some novel techniques, and leads to some new and a priori surprising results: for example the normalized error process (the difference between the estimation of σ t and the process σ t itself) is asymptotically a white noise.
Applications to the estimation of the integrated volatility are given in Sect. 13.4; in particular a thorough comparison between the methods based on multipower variations and those based on downward truncated functionals is presented.
Jean Jacod, Philip Protter

Various Extensions

Frontmatter
Chapter 14. Irregular Discretization Schemes
Abstract
In practice, observation times at stage n are quite often not regularly spaced. In this chapter, we present some results about the estimation of the integrated volatility, or more generally of integrated powers \(\int_{0}^{t}|\sigma _{s}|^{p}\,ds\), say in the one-dimensional case.
First, Sect. 14.1 presents the assumptions on the discretization schemes that are used. These assumptions cover many practical applications, but they do exclude some interesting cases, such as when the observation times are hitting times of a spatial grid by the process X.
In Sect. 14.2 we present the Law of Large Numbers for normalized functionals, possibly depending on k successive increments: the inside normalization is the square root of the length of each relevant inter-observation interval. The associated Central Limit Theorem is given in Sect. 14.3, but only for functionals depending on a single increment.
The applications to the estimation of the volatility are presented in Sect. 14.4.
Jean Jacod, Philip Protter
Chapter 15. Higher Order Limit Theorems
Abstract
In some cases, the previous Laws of Large Numbers and/or Central Limit Theorems are degenerate, in the sense that the limiting process is identically 0. In these cases, there is a need for a different normalization, which hopefully leads to a non-degenerate limit. Sect. 15.1 presents a few situations of this type.
A general theory for these cases is currently out of reach, but in Sect. 15.2 we consider a specific degenerate case, which might serve as an example for more complicated cases. Namely, we consider unnormalized functionals depending on k successive increments, in a case where the limit in the Central Limit Theorem of Chap. 11 with normalizing factor \(\sqrt {\varDelta _{n}}\) vanishes identically. We then give two different Central Limit Theorems with normalizing factor Δ n , for which the limits are non-degenerate, in two slightly different cases.
Section 15.3 is devoted to analyzing whether or not a two-dimensional process X is such that the two components have jumps at the same (random) times.
Jean Jacod, Philip Protter
Chapter 16. Semimartingales Contaminated by Noise
Abstract
In many applications, not only is the semimartingale X observed at discrete times iΔ n , but the observation is contaminated by a noise: a white noise in many “physical” applications, or a rounding noise, or a “microstructure noise” as it is called in financial econometrics.
A thorough study of the statistical problems incurred in this case would take us too far afield, but the aim of this chapter is to lay down the mathematical basis for such a study. Although there is no real application given here, the motivation is statistical applications.
The mathematical properties we are after depend fundamentally upon the structure of the noise: for example if we have a pure rounding noise there is very little one can do, and in particular there is no way to retrieve the volatility or integrated volatility. On the other hand, an additive white noise is easy to deal with, but is too restrictive for many applications, especially in financial econometrics. So we devote Sect. 16.1 to a description of the hypotheses on the noise which are necessary for our analysis. This section also contains a description of the “pre-averaging” method which we use later.
Sections 16.2 and 16.3 present the Law of Large Numbers and the associated Central Limit Theorems for the unnormalized functionals, when we plug in the increments of the pre-averaged noisy observed process instead of the those of the process X itself. The pre-averaging makes these results similar to those for functionals depending on an increasing number k n of successive increments, and the limiting processes also involve, for the Central Limit Theorem at least, the characteristics of the noise as well as those of X itself.
In Sect. 16.4 we give a Law of Large Numbers for normalized functionals, again with pre-averaging. For the Central Limit Theorem treated in Sect. 16.5, we have to restrict our attention to the one-dimensional case and for power variations, when the test function f is a linear combination of integral powers of the variable.
Finally, Sect. 16.6 is devoted to a Central Limit Theorem for the pre-averaged quadratic variation, again in the one-dimensional case, although a multi-dimensional version is also possible to reach.
Jean Jacod, Philip Protter
Backmatter
Metadata
Title
Discretization of Processes
Authors
Jean Jacod
Philip Protter
Copyright Year
2012
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-24127-7
Print ISBN
978-3-642-24126-0
DOI
https://doi.org/10.1007/978-3-642-24127-7