Optimal one-point approximation of stochastic heat equations with additive noise [Elektronische Ressource] / von Tim Carsten Wagner

Tim Wagner

Sujets

• Mathematics

Informations

• Publié par: technischen_universitat_darmstadt
• Publié le: 01 janvier 2008
• Nombre de lectures: 130
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Optimal One-Point Approximation
of Stochastic Heat Equations
with Additive Noise
Vom Fachbereich Mathematik
der Technischen Universit¨at Darmstadt
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
(Dr. rer. nat.)
genehmigte
Dissertation
von
Dipl.-Math. oec. Tim Carsten Wagner
aus Lahn-Wetzlar jetzt Wetzlar
Referent: Prof. Dr. K. Ritter
Korreferent: Prof. Dr. S. Geiß
Tag der Einreichung: 22. Juni 2007
Tag der mu¨ndlichen Pru¨fung: 30. November 2007
Darmstadt 2008
D 17Contents
Contents i
Introduction 1
1 Basic Facts 5
1.1 Deterministic Heat Equation . . . . . . . . . . . . . . . . . . . 5
1.2 Cylindrical Brownian Motion . . . . . . . . . . . . . . . . . . . 6
1.3 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Stochastic Heat Equations . . . . . . . . . . . . . . . . . . . . . 10
1.5 Wiener Sheet Approach . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 The Computational Problem 17
2.1 Approximation based on Evaluation of the Brownian Motion . 17
2.2 Classes of Algorithms . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Lower Bounds and Optimality . . . . . . . . . . . . . . 26
2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Approximation of Drift-linear SDEs with Additive Noise 31
3.1 Analysis of Minimal Errors . . . . . . . . . . . . . . . . . . . . 32
3.2 The Euler-Maruyama Scheme . . . . . . . . . . . . . . . . . . . 34
3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Optimal Approximation of Stochastic Heat Equations 51
4.1 Stochastic Heat Equations in the case B(t,x) =id . . . . . . . 51
4.2 Stochastic Heat Equations with Additive Noise . . . . . . . . . 58
4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Computational Results 77
5.1 Optimization Problems. . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Remarks on the Implementation . . . . . . . . . . . . . . . . . 82
5.3 Visualization of Realizations . . . . . . . . . . . . . . . . . . . . 83
5.4 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 89
A Numerical Results 97
iContents
List of Figures 100
List of Tables 101
Index of Notations 103
Bibliography 105
iiAcknowledgement
This thesis has been developed during my employment as Wissenschaftlicher
Mitarbeiter at the Fachbereich Mathematik of the Technische Universit¨at
Darmstadt, partially supported by the Deutsche Forschungsgemeinschaft.
I wish to express my gratitude to my adviser Prof. Dr. K. Ritter for his
valuable support during the last years. Moreover, I would like to thank Prof.
Dr. S. Geiß for being co-referee of my thesis and Prof. Dr. J. Creutzig, Prof.
Dr. M. Hieber, and Prof. Dr. P. Spellucci for being my examiners. Fur-
thermore, I would like to thank Prof. Dr. T. Mu¨ller-Gronbach and Dr. A.
Neuenkirch for fruitful discussions and inspiring comments.
I would like to thank the members of the Arbeitsgruppe Stochastik for an
motivating atmosphere during the last years.
Finally, I would like to thank my parents for their support.
iiiivIntroduction
We study approximation schemes for the mild solution of the stochastic heat
equation (
dX(t) =ΔX(t)dt+B(t)dW(t), t∈(0,T],
(0.1)
X(0) =ξ
2 don the Hilbert space H = L ((0,1) ). Here Δ denotes the Laplace operator
with Dirichlet boundary conditions, B is an operator-valued mapping and
W = (W(t)) is a (cylindrical) Brownian motion onH. The initial con-t∈[0,T]
dition ξ ∈ H is assumed to be deterministic. Note that (0.1) is a stochastic
heat equation with additive noise, since B does not depend on X(t).
bWe are interested in strong approximations X(T) of the mild solution X
at a fixed time instance t = T, and to this end we consider algorithms that
evaluate a fixed number of one-dimensional components hW,hi of W at ai
finite number of nodes t . Specifically,k,i
dY
d/2h (u) =2 sin(iπu ),i l l
l=1
so that (h ) d forms a complete orthonormal system inH, which consists ofi i∈N
beigenfunctions of Δ. The error of any approximation X(T) is defined by
1/2
2b be X(T) = EkX(T)−X(T)k , (0.2)
and its cost is defined as the total number of evaluations of the real-valued
processes hW,hi. The Nth minimal errori
n o b be(N) =inf e X(T) cost X(T) ≤N (0.3)
is the minimal error that can be achieved by any algorithm with cost at most
N. For results and references concerning minimal errors for continuous prob-
lems we refer to [TWW88, N88, R00].
Let Q denote the covariance operator of W(1). We either consider the
(ID) case, where d =1 and Q=id, or the (TC)case, whered∈N and
Qh =λ ·hi i i
with
−γλ =|i| .i 2
In the (ID) case, (0.1) is called a stochastic heat equation with space-time
white noise, while (0.1) is called a stochastic heat equation with trace class
1