Statistical inference for generalized mean reversion proces [Elektronische Ressource] / vorgelegt von Thomas Kott

Kott, Jean Thomas

Sujets

• Mathematics

Informations

• Publié par: ruhr-universitat_bochum
• Publié le: 01 janvier 2010
• Nombre de lectures: 15
• Langue: English

Extrait

Statistical Inference for Generalized
Mean Reversion Processes
vorgelegt von
Thomas Kott
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
an der Fakultät für Mathematik
der Ruhr-Universität Bochum
September 2010
Betreuer:
Prof. Dr. Herold DehlingiiContents
Introduction 1
1 Preliminaries 5
1.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Statistical Inference for Diffusion Processes . . . . . . . . . . . . . . . . . . . . 18
1.4 Linear Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Time-Discrete Observations 27
2.1 The Classical Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . 27
2.2 Method of Moments: Yule-Walker Estimator. . . . . . . . . . . . . . . . . . . . 29
2.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4um Likelihood in Practice: Numeric Search for Solutions . . . . . . . . . 42
2.5 Bias through Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 A Maximum Likelihood Approach 45
3.1 Generalized Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Consistency and Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 A Change Point Problem 63
4.1 Change in the Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Generalized Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Asymptotic Behavior of the Test Statistic . . . . . . . . . . . . . . . . . . . . . 67
4.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 How to Deal with Jumps? 79
5.1 Maximum Likelihood vs. Time-Continuous Least Squares . . . . . . . . . . . . 79
5.2 Jump Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Time-Continuous Least Squares Estimation . . . . . . . . . . . . . . . . . . . . 81
5.4 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.5 Consistency of Least Squares Estimator . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Simulation 93
Bibliography 99
iiiIntroduction
This thesis is concerned with large sample theory in statistical inference for parametric time-
inhomogeneous diffusion processes which are defined assolutionsofstochasticdifferentialequa-
tions of the form
dX =μ(t,X )dt+σdL , t≥ 0,t t t
where (L ) is a Lévy process or in particular a Brownian motion. Drawing a statisticalt t≥0
conclusion on the parameter from historical observations after a parametric model has been
chosen is one of the primary interests in practice. The outcome of such a conclusion might be
a decision about making further analysis or an adjustment of the statistical model before its
final implementation. This work concentrates on statistical inference for the parameterθ of the
drift μ(t,X ,θ) which is linear in θ. Thereby, the diffusion coefficient σ is considered knownt
and independent of θ. The incentive for this framework is stated below.
The study of inference problems for stochastic processes can be divided into two categories
in terms of the observation scheme: the considered stochastic process may be assumed to be
observed either in continuous time or at a discrete set of time points. Here, the focus of
the theory lies on the continuous time framework providing a continuous time sample path
{X ,0 ≤ t ≤ T}. The asymptotic results which are the main purpose in large sample theoryt
are obtained as T → ∞. This asymptotic concept complies with the classical setting of dis-
crete observations X ,...,X , n ∈ N, where n → ∞. The reasons justifying this continuous1 n
time setting are the following: first, there exist many real-life stochastic systems that evolve
in continuous time such that diffusion processes seem to form an adequate class to cover this
nature. In order to obtain ‘optimal’ statistical inference procedures, the theory requires time-
continuous observations which are in line with the original time-continuous model. Second,
estimators and statistics derived from time-continuous observations of diffusion processes are
natural objects to study since they often allow for a closed-form representation in terms of
stochastic integrals such that techniques from stochastic calculus can be applied. And even
though time-continuous samples are not observed in actual practice, due to an increased pro-
cessing power of up-to-date computers, simulations and statistical experiments can provide
discrete data which are sufficiently dense such that the discretization error may be neglected.
In such a case the continuous time results are valid because the statistical error of the data
prevails the discretization error. In the situation where the available data are not sufficiently
dense it might be helpful to compare the discrete time statistics with the ‘optimal’ asymptotic
properties of the continuous time statistics in order to get a better understanding of the dis-
cretization error, for example. The interesting problem of the accuracy of a discretization of a
diffusion process and its estimators has been studied by many authors and is beyond the scope
of this work. For a simple diffusion, namely for the classical Ornstein-Uhlenbeck process, the
discretization error is briefly investigated in Section 2.5. However, Monte Carlo simulations
which require an approximation of a time-continuous process by its time-discrete version and
12
of Riemann and Itô integrals by corresponding sums are presented. These simulations provide
good results for quite fairly partitions of the time interval in which the process is observed.
Throughout this work, the diffusion parameter σ is assumed to be known. This is a usual
assumption in the theory of statistical inference for the drift of time-continuously observed
diffusion processes. It can be justified by the fact that the quadratic variation of the process
can be used to compute, rather than estimate, the volatilityσ when a time-continuous sample
path is available. The challenge of the analysis of the asymptotics of drift estimators is in this
setting the time-inhomogeneous property of the process such that it is neither stationary nor
ergodic. Thatmeansthatclassicaltheoremsliketheerdogictheoremcannotbeapplieddirectly.
The essential idea of the asymptotic study is the interpretation of the stochastic process as a
sequence of random variables that take values in some function space. This method originates
from probability theory on Banach spaces.
Outline
The present dissertation is organized as follows: After the Preliminaries which summarize
some fundamental notions required for the purpose of this treatise, Chapter 2 gives an intro-
duction to the large sample theory in the classical setting of a time-discretely sampled time-
continuous stochastic process. In detail, two parameter estimation methods for the popular
mean-reverting Ornstein-Uhlenbeck process with stochastic differential
dX =α(L−X )dt+σdBt t t
are investigated. Consistency and asymptotic normality of the estimators of the parameter
t(α,L,σ) areprovedbymeansofwell-knownresultsfromthetheoryofautoregressiveprocesses
and Markov processes.
In Chapter 3 a generalized Ornstein-Uhlenbeck process defined as the solution of
dX = (L(t)−αX )dt+σdB, t≥ 0, X =ζ, (1)t t t 0
where the mean reversion function L :R →R is of the form+
p
X
L(t) = μϕ (t)i i
i=1
withknownfunctionsϕ (t),...,ϕ (t), isproposed. Thelargesamplebehaviorofthemaximum1 p
tlikelihood estimator of the parameterθ = (μ ,...μ ,α) based upon a time-continuous sample1 p
TX = {X ,0 ≤ t ≤ T} is studied. In doing so, the focus lies on a periodic mean reversiont
process of the form (1), that is
ϕ (t+ν) =ϕ (t),i i
which is a meaningful model in practice, in particular for energy commodity and electricity
data, since it captures important properties like mean reversion and seasonality. For this
periodic Ornstein-Uhlenbeck process strong consistency and asymptotic normality of the max-
imum likelihood estimator are obtained. The challenging issue of the investigation is the time-
inhomogeneity of the process, that is the time-dependence ofL(t) resulting in a process that is
neither stationary nor ergodic in the classical sense.
Chapter 4 deals with the problem of detecting a possible change in the drift parameter
tθ = (μ ,...μ ,α) of the periodic mean reversion process described above with a given sample1 p3
Tpath X ={X ,0≤t≤T} at hand. Thereby, the process (1) is written ast

0dX = S(t,X ,θ)1 +S(t,X ,θ )1 dt+σdB, 0≤t≤T,t t {t≤τ} t {t>τ} t
where
p
X
S(t,X ,θ) = μϕ (t)−αX ,t i i t
i=1
and the likelihood ratio test is proposed as a hypothesis test that decides whether or not there
0exists some τ ∈ [0,T] of the form τ = sT, s ∈ (0,1), such that θ = θ . It is shown that the
corresponding test statistic converges in distribution under the null hypothesis of no c