A Wavelet Tour of Option Pricing

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1 Universität Ulm Fakultät für Mathematik und Wirtschaftswissenschaften A Wavelet Tour of Option Pricing Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Mathematik und Wirtschaftswissenschaften der Universität Ulm vorgelegt von Roman Mario Xerxes Rometsch aus Ulm 2

2 Amtierender Dekan: Prof. Dr. Werner Kratz. Gutachter: Prof. Dr. Karsten Urban 2. Gutachter: Prof. Dr. Rüdiger Kiesel Tag der Promotion: 29. Oktober 2

3 A Wavelet Tour of Option Pricing Adaptive Wavelet Methods for Variational Inequalities 2 Mathematics Subject Classification: 35J2, 35J6, 35J86, 35K85, 35Q68, 4A25, 45E5, 49J4, 65K5, 65R2, 65T6, 9G8 c 2 Mario Rometsch Please report errors to:

4 to Sabrina

5 Abstract The efficient numerical solution of elliptic variational inequalities in general and of obstacle problems in particular is of great interest and poses a nontrivial challenge in Numerical Mathematics. Such problems occur for example after the time-discretization of the pricing problem of an American option. The algorithm EVISOLVE, that is presented in this thesis, consists of an adaptive wavelet method that allows for an efficient and reliable solution of such elliptic inequalities. Adaptivity means that the algorithm uses results to adjust itself in order to keep the convergence rate optimal also for nonsmooth solutions and to estimate the error such that a prespecified accuracy can be guaranteed. The convergence and asymptotic optimal complexity of this algorithm is shown. The arithmetic complexity is still suboptimal at the moment. Furthermore, we present Lawa, a C++-library, that consists of building blocks for the realization of adaptive wavelet methods. This software has been designed to be applied in research and education. Finally, we show how the pricing problem for American options in various Lévymodels can be solved with uniform wavelet-galerkin methods. Zusammenfassung Die effiziente numerische Lösung von elliptischen Variationsungleichungen im Allgemeinen und Hindernisproblemen im Speziellen ist ein wichtiges und nichttriviales Teilgebiet der Numerischen Mathematik und tritt z.b. als Teilproblem bei der quantitativen Bewertung Amerikanischer Optionen auf. Mit dem in dieser Dissertation vorgestellten Algorithmus EVISOLVE, der ein adaptives Wavelet-Verfahren zur Lösung von elliptischen Variationsungleichungen darstellt, können solche Ungleichungen nun effizient und verlässlich gelöst werden. Adaptiv bedeutet, dass der Algorithmus sich selbst über bereits berechnete Teilresultate anpasst. Dies geschieht mit dem Ziel, die Konvergenzgeschwindigkeit bei nicht-glatten Problemen optimal zu halten und Aussagen über die erreichte Genauigkeit zu geben, sodass eine bestimmte Fehlerschranke erreicht wird. Für diesen Algorithmus wird die Konvergenz und die asymptotisch optimale Komplexität gezeigt. Der arithmetische Aufwand ist im Moment noch suboptimal. Weiterhin wird in dieser Arbeit die C++-Bibliothek Lawa vorgestellt, die aus Bausteinen zur Realisierung adaptiver Wavelet-Verfahren besteht und mit Hinblick auf den Einsatz in Forschung und Lehre konzipiert wurde. Schließlich führen wir noch die Bewertung von Amerikanischen Optionen in verschiedenen Lévy-Modellen mit uniformen wavelet-galerkin Methoden aus.

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7 Contents. Introduction 2. Wavelets and Adaptive Methods Biorthogonal Multiresolution Analysis Biorthogonal Spline Multiresolution on the Real Line Properties of Biorthogonal Spline Wavelets Biorthogonal Spline Wavelets on the Interval Adaptive Wavelet Methods for Elliptic Equations Prerequisites Elliptic Operator Equations Uniform Schemes Best N-term Approximation Besov Regularity for Elliptic Equations in One Dimension Abstract Adaptive Method Prerequisites for an Implementable Adaptive Method Adaptive Wavelet Solver for Elliptic Equations Adaptive Wavelet Solver for Elliptic Equations without Coarsening the Iterands A Simplified Adaptive Wavelet Solver for Elliptic Equations Adaptive Wavelet Methods for Nonlinear Equations Nemytskij Operators and their Properties Recovery Scheme Recovery-based Approximate Matrix-vector Product Solving Operator Equations with Lawa Operator Equations Helmholtz Equation Hypersingular Integral Equation Model Problems that come with Lawa Examples for the Helmholtz Problem Examples for the Hypersingular Problem Uniform Schemes Best N-term Approximation Adaptive Wavelet Methods ELLSOLVE ELLSOLVE-WO-COARSENING S-ADWAV-ELLSOLVE Semilinear Equations Semilinear Helmholtz Example vii

8 Contents Uniform Methods Adaptive Wavelet Methods Adaptive Wavelet Solution to Variational Inequalities Prerequisites Elliptic Variational Inequalities Parabolic Variational Inequalities Uniform Methods for EVIs and PVIs EVIs PVIs Adaptive Methods for EVIs Besov Regularity for EVIs Abstract Adaptive Method Adaptive Projection Numerical Tests for PROJECT EVISOLVE S-ADWAV-EVISOLVE Adaptive Methods for PVIs Reformulation as an Operator Inequality Formulation as a Bi-infinite Inequality in Wavelet Space Towards PVISOLVE Solving Inequalities with Lawa Elliptic Obstacle Problems Second-order Differential Operator Examples for the Helmholtz Inequality Uniform Schemes On Residual Estimation for Elliptic Obstacle Problems Convergence Rates of Uniform Schemes Best N-term Approximation EVISOLVE Leaving the Inner Loop with the Exact Residual Leaving the Inner Loop with the Approximate Residual Leaving the Inner Loop with the Simplified Residual S-ADWAV-EVISOLVE Lévy Processes & American Options Options and their Fair Value Lévy Processes The Geometric Brownian Motion The Variance Gamma Process The Normal Inverse Gaussian Process The CGMY Process The Merton Jump-diffusion The Generalized Hyperbolic Lévy Motion The Kou Model Fully Deterministic Option Pricing viii

9 Contents 7. Setup of the Algebraic System and Singular Quadrature Hadamard Finite-Part Integral General Calculation Specific Implementation The CGMY Kernel The Kou Jump-diffusion The Merton Jump-diffusion The Normal Inverse Gaussian Model Pricing Options with Lawa European Options Black-Scholes Model Merton Jump-diffusion Model Kou Jump-diffusion Model CGMY Pure-jump Model CGMYe Model Normal Inverse Gaussian Model American Options Black-Scholes Model Merton Jump-diffusion Model Kou Jump-diffusion Model CGMY Pure-jump Model CGMYe Model Normal Inverse Gaussian Model Further Models and Option Types Conclusion 79.Outlook 8 A. Stochastics 85 B. Function Spaces 87 B.. Sequence Spaces B.2. Lebesgue and Sobolev Spaces B.3. Vector-valued Lebesgue and Sobolev Spaces B.4. Nonlinear Approximation Spaces C. Singular Quadrature 97 C.. Gaussian Quadrature C.2. Computation of Gaussian Rules C.3. Gaussian Finite-Part Quadrature D. Complexity Analysis for Quadratic Programming 2 ix

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11 List of Figures 2.. Centralized B-Splines of order d =,..., Dual B-Splines Ñ,,..., Ñ4, Wavelet-based image compression Wavelet preconditioning Primal [DKU99] wavelets on the interval The approximation of a function by wavelets Slopes for APPLY Multiscale and local scaling functions representation Slopes for RECOVERY BU and F, d = d = Slopes for RECOVERY BU and F 2, d = d = Slopes for RECOVERY CDD and F, d = d = Slopes for RECOVERY CDD and F 2, d = d = Slopes for RECOVERY CDD and F, d = d = Slopes for RECOVERY CDD and F 2, d = d = Slopes for RECOVERY APPLY CDD and A, d = d = Slopes for RECOVERY APPLY CDD and A, d = d = Slopes for RECOVERY APPLY CDD and A 2, d = d = Plots of the examples for the Helmholtz problem (P) and (P2) Plots of the examples for the Helmholtz problem (P3) and (P4) Plots of the examples for the hypersingular problem (P) and (P2) Convergence rates of uniform schemes for the Helmholtz problem Convergence rates of uniform schemes for the hypersingular problem Convergence rates of best N-term approximation for the Helmholtz examples Convergence rates of best N-term approximation for the hypersingular examples ELLSOLVE, Helmholtz problem (P3) ELLSOLVE, Helmholtz problem (P4) ELLSOLVE, hypersingular problem (P2) ELLSOLVE-WO-COARSENING, Helmholtz problem (P3) ELLSOLVE-WO-COARSENING, Helmholtz problem (P4) ELLSOLVE-WO-COARSENING, hypersingular problem (P2) S-ADWAV-ELLSOLVE, Helmholtz problem (P3) S-ADWAV-ELLSOLVE, Helmholtz problem (P4) S-ADWAV-ELLSOLVE, hypersingular problem (P2) Residual reduction rates of the NEWTON solver xi

12 List of Figures 3.8. Convergence rates of uniform schemes for the semilinear Helmholtz problem ELLSOLVE-WO-COARSENING, semilinear Helmholtz problem (P3) ELLSOLVE-WO-COARSENING, semilinear Helmholtz problem (P4) PROJECT BU, Helmholtz inequality (P) and (P2) PROJECT BU, Helmholtz inequality (P3) and (P4) PROJECT BU, Helmholtz inequality (P5) and (P6) PROJECT CDD, Helmholtz inequality (P) PROJECT CDD, Helmholtz inequality (P2) PROJECT CDD, Helmholtz inequality (P3) PROJECT CDD, Helmholtz inequality (P4) PROJECT CDD, Helmholtz inequality (P5) PROJECT CDD, Helmholtz inequality (P6) Plots of the examples for the Helmholtz inequality problem Plots of the homogenized examples for the Helmholtz inequality problem Error estimation for the projected Richardson iteration, J = Error estimation for the projected Richardson iteration, J = Error estimation for the projected Richardson iteration, J = Convergence rates of uniform schemes for the Helmholtz inequality problem Convergence rates of best N-term approximation for the Helmholtz inequality problem EVISOLVE for the Helmholtz inequality problem (P4) EVISOLVE for the Helmholtz inequality problem (P5) EVISOLVE for the Helmholtz Inequality problem (P6) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P4) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P5) EVISOLVE using the approximate residual for the Helmholtz inequality problem (P6) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P4) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P5) EVISOLVE using the simplified residual for the Helmholtz inequality problem (P6) S-ADWAV-EVISOLVE for the Helmholtz inequality (P) S-ADWAV-EVISOLVE for the Helmholtz inequality (P2) S-ADWAV-EVISOLVE for the Helmholtz inequality (P3) S-ADWAV-EVISOLVE for the Helmholtz inequality (P4) S-ADWAV-EVISOLVE for the Helmholtz inequality (P5).. 35 xii

13 List of Figures S-ADWAV-EVISOLVE for the Helmholtz inequality (P6) Sample paths of different stock price models Smooth pasting principle in the Black-Scholes model Violated smooth pasting principle in the CGMY model European options in the Black-Scholes model European options in the Merton jump-diffusion model European options in the Kou jump-diffusion model European options in the CGMY model European options in the CGMYe model European options in the Normal Inverse Gaussian model American options in the Black-Scholes model American options in the Merton Jump-diffusion model American options in the Kou Jump-diffusion model American options in the CGMY model American options in the CGMYe model American options in the Normal Inverse Gaussian model xiii

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15 List of Algorithms. COARSE[η, v] v η RHS[η, f] f η APPLY[η, A, v] w η GALSOLVE[Λ, A Λ Λ, ṽ Λ, f Λ, η] v Λ ELLSOLVE[ε, A, f] u(ε) ELLSOLVE-WO-COARSENING[ε, A, f] u(ε) C(Λ, c) Λ S-ADWAV-ELLSOLVE[ε, A, f] u(ε) RECOVERY[η, F, u Λ ] vˆλ PREDICTION[ε, u Λ ] Γ RECONSTRUCTION[Γ, u Λ ] v Γ QUASI-INTERPOLATION[v Γ, F] g CHANGE-OF-BASIS[g] g DECOMPOSITION[u] v PREDICTION CDD[ε, u Λ ] Γ DECOMPOSITION [u] v NEWTON[Λ, A Λ Λ, F, ṽ Λ, f Λ, η] v Λ PRICHARDSON[Λ, A Λ Λ, ũ Λ, f Λ, η] u Λ PPRICHARDSON[Λ, A Λ Λ, ũ Λ, f Λ, η] u Λ PROJECT LSF[η, ˆv] w η PROJECT[η, ˆv] w η EVISOLVE[ε, A, f] u(ε) EXTRAPOLATE[u Λ ] ˆΠ(u Λ ) S-ADWAV-EVISOLVE[ε, A, f] u(ε) xv

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17 . Introduction... and these wavelets, which science burst upon the world that night, were strange even to the women and men who used them. We start this thesis with a short introduction, where we present the object of analysis, namely the elliptic and the parabolic variational inequality. In order to relate the adaptive methods developed in this thesis later on we give a short overview on the various numerical methods for solving these inequalities. A survey on wavelet methods and their application to computational finance follows together with a comparison of other numerical schemes for the option pricing problem. We close this chapter with an outline concerning the adaptive wavelet library Lawa and a summary of the contributions of this thesis. The Problem To give an idea for the problems analyzed in this thesis, the next figure shows two important applications of variational inequalities, namely the elliptic obstacle problem on the left and the American option pricing problem on the right side Membrane Obstacle V (t, S) Calculated Put value Payoff S t Both problems have in common that they are partially driven by some sort of operator equation, but only in a sub-domain. In the other part of the domain the solution rests on the obstacle. It is also known what happens on the boundary of these sub-domains, but the important question now is where this boundary is located this is why such problems are also called free boundary problems. The important starting point with existence and uniqueness results for variational inequalities was laid in [LS67]. Since then, a whole field of research has emerged concerning the efficient solution of these problems, from which adaptive methods are nowadays among the most advanced schemes.

18 . Introduction Adaptive Finite Element Methods for Inequalities In this context adaptivity means that schemes use estimates to adapt themselves to the local behaviour of the solution of the problem in order to use a certain budget of computing time in a most efficient way. Adaptive methods for solving variational inequalities in general and obstacle problems in particular are nowadays well-established in the form of adaptive finite element methods (AFEM). Error estimators of residual type for elliptic obstacle problems in connection with linear finite elements are derived in [CN, Vee2]. In [Bra5] the author shows that an error estimator based on the Lagrange multiplier is also reliable and efficient but gives rise to a nonconforming contribution. The convergence of a conforming AFEM for obstacle problems again based on the Laplace operator is established in [BCH7] for affine obstacles. However, the presented method attains the same convergence rate as a uniform scheme. Similar findings are presented in [BCH9] for general obstacles where again the adaptive strategy yields only the convergence rate of a uniform scheme while displaying a slightly smaller absolute error. In [BC4] the authors derive error estimators for obstacle problems for the Laplace operator based on averaging techniques. If one is interested in bounding the error in the maximum norm then the estimators from [NSV3] are a good choice as they are reliable and efficient as well, which is demonstrated with the help of examples in 2d and 3d. For parabolic inequalities driven by the Black-Scholes operator, the authors show in [MNvPZ7] that their error estimator is reliable and efficient and gives rise to higher convergence rates. The authors also apply the resulting adaptive method to the example of American option pricing. For an extension to inequalities of parabolic and elliptic type that are driven by integro-differential operators we refer to [NvPZ8]. There, a posteriori error estimates for piecewise linear finite elements are derived and tested at various examples. When it comes to the solution of the discrete systems that arise during an AFEM computation the algorithms from [Hop87] or [Kor94, Kor96] may be used as they are globally convergent and have the same optimal complexity as standard multi-grid methods for elliptic equations. Wavelets Compactly supported bases of orthonormal or biorthogonal wavelets are known since [Dau88, CDF92]. Since then, these functions have been used in applications as diverse as for example image procession, audio denoising, data compression and the solution of PDEs. Following the discovery of wavelet bases on the real line, the next step towards wavelet bases on general domains is the construction on the interval [, ] that was done in [DKU99, Pri6, Dij9] to name only a few. Wavelet bases for general domains can be constructed via domain decomposition and have been proposed for example in [CTU99, CTU, BE2, CM, DS99]. It is more or less straight-forward how to use wavelets for the solution of operator equations. Later on, still several other advantages like wavelet preconditioning 2

19 [DK92, Jaf92] or wavelet-based operator compression were discovered [BCR9, DPS93, Sch98, Ste4, GS6, GS5]. Starting from [CDD], various contributions have been made to adaptive wavelet methods in [CDD2, GHS7, Gan8] for elliptic equations. Proven convergence rates for AFEM schemes are also available as it was shown in [BDD4]. Adaptive methods for nonlinear problems have been analyzed in [DSX, CDD3, CDD4, BU8]. In [DHU, DDU2] optimal adaptive schemes for the solution of saddle point problems like Stokes problem have been developed. There are also specialized wavelet bases and schemes available for the Stokes problem, see [Urb95a, Urb, Urb95b, Urb96, DKU96]. Further applications of adaptive wavelet methods are for example the adaptive optimization of convex functionals in Banach spaces that was presented in [CU5], where this list is far from being complete. Wavelet Methods for Option Pricing Starting from [MvPS4], where wavelet-galerkin methods were applied for the first time to the pricing of European options, there has been a tremendous progress in this particular field of research. Since then, many other subproblems from quantitative finance have been analyzed like more sophisticated time-stepping for European options [MSW5, MSW6], American option pricing [MNS5], Swing option valuation [WW8], models incorporating stochastic volatility [HMS5, Hil9], variational sensitivity analysis [HSW8], varianceoptimal hedging [Ves9] or multidimensional Lévy models [FRS7, Rei8b, Rei8a, LRZ8, WR8, Win9, RSW9, Rei], where this list does not claim to be complete. See also the survey article [HRSW9]. The simple nature of the domain of problems arising in option pricing, which are essentially cubes, plays in the hands of wavelet methods as there the construction of the bases is straight-forward. Finite Differences and Monte-Carlo Option Pricing Finite difference methods (FDM) for partial integro-differential equations (PIDE) arising in option pricing were presented in [CV6]. As these FD methods are not able to treat the singular integral operator directly, the singularity of the Lévy kernel has to be regularized by truncating the jump measure in some small neighbourhood of the origin and adding a corresponding diffusion component to the infinitesimal generator of the process. While this is justified for European options, in [Pow9] it is found that this is not appropriate for exotic contracts like American options or Barrier contracts. In [KL9] the author also mentions that the same effect is true for Monte-Carlo methods. Therefore, finite element methods, which do not require regularization, seem to be the natural method of choice for that type of problems. 3

20 . Introduction A Library for Adaptive Wavelet Applications All code that has been developed during the creation of this work is publicly available, see [RS, RSU]. The reason for that is manifold: firstly, we strongly believe in the importance of the underlying code for the validity of the numerical experiments conducted in this thesis that back the theoretical findings. As found in [Hat97], many scientific calculation codes contain serious flaws. For this issue, which has just recently been raised in [Inc] and then extensively discussed in [Sou], the availability of code used in scientific publications is essential for other researchers in that academic field to validate the results of a specific paper. Secondly, the accessibility of work, that has already been accomplished also greatly saves time for people who work in that particular area (the often exerted reinvention of the wheel) or especially who are starting research in that domain. Lastly, every researcher can ask himself the question about the worthiness of numerical results when no one knows how they have been produced. As the algorithms for wavelet-based methods are quite involved and this particular field of research is also very young, there are no matured software projects or even commercial packages available. A first attempt to resolve this drawback was done in the IgpmLib as described in [BBJ + 2]. However, this library has some slight design flaws, which mainly originate from the poor compiler compliance to the C++-standard back then and the fact that the library was in the first place not intended for adaptive methods. Therefore, back in 25 Alexander Stippler came to the decision that only a complete rewrite from scratch would solve this problem and he began developing the library Lawa, which is an abbreviation of Library for Adaptive Wavelet Applications within his Ph.D.-Project [Sti]. The author of this thesis joined this project in 27, which is publicly available at under the terms of the GNU General Public License [Fre9, Version 2] or [Fre7, Version 3]. Of course, the library is still under development and thus far from being complete nor free from defects and so this thesis can only represent a snap-shot of the methods currently available in Lawa. However, in making it a public project the authors hope to encourage other researcher on working with that foundation. Contributions The contributions of this thesis are the design of adaptive wavelet algorithms for the solution of variational inequalities and the development of a modern replacement of the IgpmLib in view of a library for adaptive wavelet applications that is free as in free speech, portable and usable in research and education. Starting in Chapter 2, we provide a short introduction to wavelet and multiscale methods with a focus on the adaptive wavelet algorithms from [CDD, CDD2, GHS7, BK6]. We also present one of the first implementations of the RECOVERY scheme, which approximatively evaluates nonlinear 4

21 functions of wavelet expansions, as developed in [BU8, CDD4] including some numerical tests. Then in Chapter 3, we introduce two different model problems for an operator equation that come with Lawa and which can be solved by the adaptive algorithms from the second chapter. Building on a slightly different variant of the RECOVERY scheme, we design in Chapter 4 an adaptive projection algorithm and based on that a first adaptive wavelet algorithm EVI- SOLVE for solving elliptic variational inequalities, that is fully computable with proven convergence. As of today, there are no previous adaptive wavelet methods for variational inequalities known to the author. Besides analysing the Besov regularity of the elliptic variational inequality, a second simplified adaptive wavelet algorithm S-ADWAV-EVISOLVE is presented. Finally, a Galerkin solver for the resulting finite-dimensional variational inequalities is derived and a heuristic residual-based error bound is proposed, which is also tested at some examples. We also present some building blocks towards an adaptive wavelet scheme for parabolic inequalities. In Chapter 5 the two adaptive algorithms are applied to a set of both smooth and singular elliptic test examples. When we come to the quantitative finance part of this thesis in Chapter 6, the gap between the American option pricing problem and parabolic variational inequalities is bridged by recalling major theorems from [MvPS4, MNS5]. We also present various different Lévy models that are commonly used. In Chapter 7 we apply the concept of the Hadamard finite part integral to the setup of the algebraic system stemming from the discretized option pricing problem. Lastly, we show in Chapter 8 how uniform schemes may be applied to the pricing problem and plot the value functions of European and American options in the various models from Chapter 6. We summarise the findings of this thesis in Chapter 9 and provide in Chapter some starting points for further research in this field that are now imminent. Miscellaneous Throughout this thesis, a b will always mean a Cb where the constant C R = { x R x } does not depend on any other quantity in the statement unless explicitly mentioned. Conversely, b a is short for b Ca and finally a b stands for a b and b a. We will always denote functions in italics u(x) while bold-faced characters indicate (infinite-)dimensional vectors u l 2 or matrices A : l 2 l 2. For a vector u l 2, we denote by u Λ its restriction to the index set Λ while we use the expression u Λ in order to point out its support Λ. The characters V and H always denote real Hilbert spaces, where as usual V and H stand for their topological duals. Typical examples used in this thesis are V = H (Ω) and H = L 2(Ω). 5

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23 2. Wavelets and Adaptive Methods All this time the Guard was looking at her, first through a telescope, then through a microscope, and then through an opera-glass. (Lewis Carroll) The first chapter now introduces the main tool that we are going to use in the course of this thesis for solving operator equations and inequalities. For this we will loosely follow the book [Urb9] and the survey article [Dah97], where also many proofs of the various statements may be found. The interested reader who wants to gain additional insight in the history of wavelets is also referred to those. Afterwards the adaptive wavelet methods for the solution of operator equations from [CDD, CDD2, GHS7, BK6] are described as they are implemented in Lawa and because they form the starting point for our development of the adaptive methods for operator inequalities later on. At the end of this chapter we have a look at the RECOVERY scheme that allows to evaluate nonlinear operators in an efficient way using a wavelet basis. There we will also present some experiments concerning the quantitative behaviour of the methods from [BU8] and [CDD4]. 2.. Biorthogonal Multiresolution Analysis We begin with a basic concept. Definition 2. (Biorthogonal Multiresolution Analysis) For the Hilbert space L 2 (Ω) from Definition B.7, Ω R, with inner product (, ) = (, ) ;Ω and norm = ;Ω we are considering two sequences of closed, nested subspaces {S j } j N and { S j } j N such that their union is dense in L 2 (Ω): ( ) S j S j+, clos L2 (Ω) S j = L 2 (Ω), S j S j+, clos L2 (Ω) j= ( j= S j ) = L 2 (Ω). We assume that the spaces S j and S j are generated by the basis functions ϕ j,k and ϕ j,k, respectively, which means that S j = S(Φ j ) = span(φ j ), with Φ j = { ϕj,k k I j }, where Ij Z denotes the index set of the basis functions on level j. If then for j N it holds that {, k = l, (ϕ j,k, ϕ j,l ) =, otherwise, ϕ j,k S j, ϕ j,l S j, (2.) 7

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