Weak approximation of stochastic delay [Elektronische Ressource] : differential equations with bounded memory by discrete time series / von Robert Lorenz

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Weak Approximation of Stochastic DelayDifferential Equations with Bounded Memory byDiscrete Time SeriesDISSERTATIONzur Erlangung des akademischen Gradesdoctor rerum naturalium(Dr. rer. nat.)im Fach Mathematikeingereicht an derMathematisch-Naturwissenschaftlichen Fakultät IIder Humboldt-Universität zu BerlinvonHerrn Dipl.-Math.RobertLorenzgeboren am 9. Juni 1973 in Schwedt/OderPräsident der Humboldt-Universität zu Berlin:Prof. Dr. Hans Jürgen PrömelDekan der Mathematisch-Naturwissenschaftlichen Fakultät II:Prof. Dr. Uwe KüchlerGutachter:1. Prof. Dr. Uwe Küchler2. Prof. Dr. Evelyn Buckwar3. Prof. Dr. Hans-Michael Dietzeingereicht am: 25. Oktober 2005Tag der mündlichen Prüfung: 20. März 2006AbstractConsider the stochastic delay differential equation (SDDE) with length of memory rdX(t) =b(X )dt+σ(X )dB(t),t twhich has a unique weak solution. Here B is a Brownian motion, b and σ are con-tinuous, locally bounded functions defined on the space C[−r,0], and X denotes thetsegment of the values of X(u) for time points u in the interval [t,t−r]. Our aimh his to construct a sequence of discrete time series X of higher order, such that Xconverges weakly to the solution X of the stochastic differential delay equation as htends to zero.On the other hand we shall establish under which conditions a given sequence ofhtime seriesX of higher order converges weakly to the weak solutionX of a stochasticdifferential delay equation.

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Publié le 01 janvier 2006
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Weak Approximation of Stochastic Delay
Differential Equations with Bounded Memory by
Discrete Time Series
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät II
der Humboldt-Universität zu Berlin
von
Herrn Dipl.-Math.RobertLorenz
geboren am 9. Juni 1973 in Schwedt/Oder
Präsident der Humboldt-Universität zu Berlin:
Prof. Dr. Hans Jürgen Prömel
Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II:
Prof. Dr. Uwe Küchler
Gutachter:
1. Prof. Dr. Uwe Küchler
2. Prof. Dr. Evelyn Buckwar
3. Prof. Dr. Hans-Michael Dietz
eingereicht am: 25. Oktober 2005
Tag der mündlichen Prüfung: 20. März 2006Abstract
Consider the stochastic delay differential equation (SDDE) with length of memory r
dX(t) =b(X )dt+σ(X )dB(t),t t
which has a unique weak solution. Here B is a Brownian motion, b and σ are con-
tinuous, locally bounded functions defined on the space C[−r,0], and X denotes thet
segment of the values of X(u) for time points u in the interval [t,t−r]. Our aim
h his to construct a sequence of discrete time series X of higher order, such that X
converges weakly to the solution X of the stochastic differential delay equation as h
tends to zero.
On the other hand we shall establish under which conditions a given sequence of
htime seriesX of higher order converges weakly to the weak solutionX of a stochastic
differential delay equation.
As an illustration we shall derive a weak limit of a sequence of GARCH processes
ofhigherorder. Thislimittendsouttobetheweaksolutionofastochasticdifferential
delay equation.
Keywords:
stochastic delay differential equations, weak approximation, discrete time series,
GARCH processesZusammenfassung
WirbetrachtendiestochastischeDifferentialgleichungmitGedächtnis(SDDE)mit
Gedächtnislänge r
dX(t) =b(X )dt+σ(X )dB(t)t t
mit eindeutiger schwacher Lösung . Dabei ist B eine Brownsche Bewegung, b und
σ sind stetige, lokal beschränkte Funktionen mit Definitionsbereich C[−r,0], und Xt
bezeichnet das Segment der Werte von X(u) für Zeitpunkte u im Intervall [t,t−r].
hUnser Ziel ist eine Folge von diskreten ZeitreihenX höherer Ordung zu konstruieren,
hsodassmithgegen0dieZeitreihenX schwachgegendieLösungX derstochastischen
Differentialgleichung mit Gedächtnis konvergieren.
Desweiteren werden wir Bedingungen angeben, unter denen eine gegeben Folge
hvon Zeitreihen X höherer Ordung schwach gegen die Lösung X einer stochastischen
Differentialgleichung mit Gedächtnis konvergiert.
Als ein Beispiel werden wir den schwachen Grenzwert einer Folge von diskreten
GARCH-Prozessen höherer Ordnung ermitteln. Dieser Grenzwert wird sich als schwa-
che Lösung einer stochastischen Differentialgleichung mit Gedächtnis herausstellen.
Schlagwörter:
stochastische Differentialgleichungen mit Gedächtnis, schwache Approximation,
diskrete Zeitreihen, GARCH-ProzesseContents
1 Introduction 1
2 Stochastic Delay Differential Equations Driven by a Brownian Mo-
tion 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Martingale Problem on C[−r,∞). . . . . . . . . . . . . . . . . . . 6
2.3 The in Discrete Time . . . . . . . . . . . . . . . . 10
2.4 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Approximation of a Given Stochastic Delay Diff. Equation . . . 35
2.5.2 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . 46
2.5.3 A Continuous GARCH(p,1)-Model . . . . . . . . . . . . . . . . 50
2.5.4 A Continuous GARq)-Model . . . . . . . . . . . . . . . . 56
2.5.5 Time Series with Fading Memory . . . . . . . . . . . . . . . . . 58
2.5.6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Solutions of Stochastic Delay Diff. Equations as Semimartingales . . . 64
2.7 Comparison to Literature . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Weak Limits of ARMA-Series 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Establishing the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Discussion of the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4 Comparison to Literature . . . . . . . . . . . . . . . . . . . . . . . . . 103
Bibliography 107
A Acknowledgement 111
viiChapter 1
Introduction
Convergence of stochastic processes
Let X := (X(t),t ≥ 0) be a continuous stochastic process. Often one is inter-
ested in the distribution of certain functionals of the process, for instance of φ(X) =
max X(t). In general it is difficult to determine the distribution of φ(X). One0≤t≤T
way to tackle this problem is to consider an appropriate sequence of processes Xn
converging weakly toX. Sometimes the distribution of the functional under consider-
ation can be determined much easier for every X , and the distribution for X can ben
obtained as a limit distribution. This is exactly the procedure of Donsker’s invariance
principle. Along this line we shall establish convergence results for weak solutions of
stochastic delay differential equations.
Stochastic delay differential equations
Stochastic delay differential equations (SDDE’s) have become widespread in the last
30 years. Phenomena of time delay occur in many different areas of the real world.
Stochasticdelaydifferentialequationsaretheirmathematicalreflection. Adescription
without time delay is nowadays not to think of. In physics it is the time of transporta-
tion of particles or information from one system to another. In financial mathematics
it is the time to react on developments in financial markets. In econometrics time
delay corresponds to the reaction time of the client to behave in a certain way. A first
survey on the theory of SDDE’s is presented in Mohammed [21] and Mao [19]. We
shall consider SDDE’s of the kind

X(t) = ξ(t), −r≤t≤ 0
(1.0.1)
dX(t) = b(X )dt+σ(X )dB(t), t≥ 0.t t
Here B is a Brownian motion, b and σ are measurable, locally bounded functions de-
finedonC[−r,0],andξ isadeterministic, continuousfunctionon [−r,0]. Furthermore
X is the segment (X(t+u)) , where r≥ 0 denotes the length of memory. Wet −r≤u≤0
exclude the case r =∞ which has been treated in Riedle [25]. In the case r = 0 the
12 Introduction
system (1.0.1) is a stochastic ordinary differential equation. The aim of this work is to
approximateweaklythesolutionof(1.0.1). WeshallconstructprocessesX whichwilln
converge weakly to the processX, whereX is the unique weak solution of the system
(1.0.1). Thiswillbedonefirstlyifthecoefficientsbandσ arecontinuousandbounded,
secondly if they are continuous and locally bounded, finally we admit discontinuity
points for the coefficients b and σ. The approximating processes are constructed in a
(h)
first step as autoregressive time series (X ) on a time grid {mh : m∈N } form∈N 00mh
(h)h > 0. The quantity h is called step length. In a second step X is extended to a
continuous process by linear interpolation. To indicate the correspondence to the step
(h)length h, we shall denote the approximating processes by X rather than by X .n
History of convergence results for stochastic processes
One of the first results on convergence of stochastic processes is the famous Donsker
theorem, see Billingsley [2], Theorem 10.1. If { } is a sequence of i.i.d. centeredk
random variables with variance 1, then the sequence of processes defined by
[nt]
X1
S (t) := √ , 0≤t≤Tn k
n
k=1
converges weakly to a Brownian motion on [0,T] asn tends to infinity. The Brownian
motion is a special case of a Markov diffusion with vanishing drift coefficient and
diffusion coefficient 1.
A general result on convergence of stochastic processes to a Markov diffusion is
presentedinStroockandVaradhan[28]. Foreachh> 0letusbegivenad-dimensional
(h) (h) (h)(h)Markov chain X = (X ,X ,X ,...) with transition probabilities0 h 2h
(h) (h) (h) (h) dp (X ,A) =P(X ∈A|X ), A∈B . (1.0.2)kh kh kh(k+1)h
In terms of the transition probabilities the following quantities are defined for t≥ 0
dand x∈R
Z
1 (h)(h)a (t,x) := (y−x)p (x, dy)t[ ]h
hh d
R
Z
1 (h)(h) (h)2 Tb (t,x) =σ (t,x) := (y−x)(y−x) p (x, dy).t[ ]h
hh d
R
(h)A stochastic processX (t) in continuous time is constructed by linear interpolation.
If there exist functions a and σ such that
(h) (h)2 2 da (t,x)−−→a(t,x), σ (t,x)−−→σ (t,x), t≥ 0, x∈R
h→0 h→0
d (h)uniformlyoncompactsetsofR ×R ,thenthesequenceofprocessesX (t)converges+
weakly to a Markov diffusionX with coefficientsa andσ. This means, the processX