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104 CHAPTER 3. NUMERICAL IMPLEMENTATION completely blurred and the data does not allow to construct a better approximation of the true solution than Q ∗ . Let Q δ α denote a solution of the regularized problem (2.27) with data C δ M parameter α. The function and regularization ε δ (α) := ‖C Qδ α − C δ M‖ 2 w is called the discrepancy function of the calibration problem (2.27). Note that since this problem can have many solutions, ε δ (α) is a priori a multivalued function. Given two constants c 1 and c 2 satisfying the discrepancy principle can be stated as follows: 1 < c 1 ≤ c 2 < εmax δ0 2 , (3.12) Discrepancy principle For a given noise level δ, choose α > 0 that satisfies δ 2 < c 1 δ 2 ≤ ε δ (α) ≤ c 2 δ 2 , (3.13) If, for a given α, the discrepancy function has several possible values, the above inequalities must be satisfied by each one of them. The intuition behind this principle is as follows. We would like to find a solution Q of the equation C Q = C M . Since the error level in the data is of order δ, it is the best possible precision that we can ask for in this context, so it does not make sense to calibrate the noisy data CM δ with a precision higher than δ. Therefore, we try to solve ‖C Qδ α − C δ M ‖ 2 w ≤ δ 2 . In order to gain stability we must sacrifice some precision compared to δ, therefore, we choose a constant c with 1 c, for example, c = 1.1 and look for Q δ α in the level set ‖C Qδ α − CM‖ δ 2 w ≤ cδ 2 . (3.14) Since, on the other hand, by increasing precision, we decrease the stability, the highest stability is obtained when the inequality in (3.14) is replaced by equality and we obtain ε δ (α) ≡ ‖C Qδ α − CM‖ δ 2 w = cδ 2 . To make the numerical solution of the equation easier, we do not impose a strict value of the discrepancy function but allow it to lie between two bounds, obtaining (3.13).
3.3. CHOICE OF THE REGULARIZATION PARAMETER 105 Supposing that an α solving (3.13) exists, it is easy to prove the convergence of the regularized solutions to the minimal entropy least squares solution, when the regularization parameter is chosen using the discrepancy principle. Proposition 3.3. Suppose that the hypotheses 1–3 of page 103 are satisfied and let c 1 and c 2 be as in (3.12). Let {C δ k M } k≥1 be a sequence of data sets such that ‖C M − C δ k M ‖ w < δ k and δ k → 0. Then the sequence {Q δ k αk } k≥1 where Q δ k αk is a solution of problem (2.27) with data C δ k M , prior P and regularization parameter α k chosen according to the discrepancy principle, has a weakly convergent subsequence. The limit of every such subsequence of {Q δ k αk } k≥1 is a MELSS with data C M and prior P . Proof. Using the optimality of Q δk α k , we can write: ε δk (α k ) + α k I(Q δk α k |P ) ≤ ‖C Q+ − C δ k M ‖2 w + α k I(Q + |P ) ≤ δ 2 k + α kI(Q + |P ). The discrepancy principle (3.13) then implies that I(Q δk α k |P ) ≤ I(Q + |P ), (3.15) By Lemma 2.10, the sequence {Q δ k αk } k≥1 is tight and therefore, by Prohorov’s theorem and Lemma 2.4, relatively weakly compact in M ∩ L + B . Choose a subsequence of {Q δ k αk } k≥1 , converging weakly to a limit Q and denoted, to simplify notation, again by {Q δ k αk } k≥1 . Inequality (3.13) and the triangle inequality yield: ‖C Qδ k αk − C M ‖ w ≤ ‖C Qδ k αk − C δ k M ‖ w + δ k ≤ δ k (1 + √ c 2 ) −−−→ k→∞ 0. By Lemma 2.2 this means that ‖C Q − C M ‖ w = 0 and therefore Q is a solution. By weak lower semicontinuity of I (cf. Corollary 2.1) and using (3.15), I(Q|P ) ≤ lim inf k which means that Q is a MELSS. I(Q δ k αk |P ) ≤ lim sup k I(Q δ k αk |P ) ≤ I(Q + |P ), The discrepancy principle performs well for regularizing linear operators but may fail in nonlinear problems like (2.27), because Equation (3.13) may have no solution due to discontinuity of the discrepancy function ε δ (α). Examples of nonlinear ill-posed problems to which the
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Thèse présentée pour obtenir le
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Abstract This thesis deals with the
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Contents Remarks on notation 11 Int
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CONTENTS 9 II Multidimensional mode
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12 For {P n } n≥1 , P ∈ P(Ω) t
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14 INTRODUCTION EN FRANCAIS corresp
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16 INTRODUCTION EN FRANCAIS Lévy p
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18 INTRODUCTION EN FRANCAIS Calibra
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20 INTRODUCTION EN FRANCAIS • Si
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22 INTRODUCTION EN FRANCAIS et Scai
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24 INTRODUCTION EN FRANCAIS Ce rés
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26 INTRODUCTION EN FRANCAIS dans un
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28 INTRODUCTION 7500 Taux de change
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30 INTRODUCTION Lévy models by Fou
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Chapter 1 Lévy processes and exp-L
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1.1. LEVY PROCESSES 35 Proposition
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1.1. LEVY PROCESSES 37 The characte
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1.1. LEVY PROCESSES 39 On the other
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1.2. EXPONENTIAL LEVY MODELS 41 (b)
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1.3. EXP-LEVY MODELS IN FINANCE 43
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1.3. EXP-LEVY MODELS IN FINANCE 45
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1.4. PRICING EUROPEAN OPTIONS 47 wi
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1.4. PRICING EUROPEAN OPTIONS 49 Pr
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1.4. PRICING EUROPEAN OPTIONS 51 Nu
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154 CHAPTER 4. DEPENDENCE OF LEVY P
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166 CHAPTER 5. APPLICATIONS OF LEVY
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188 CONCLUSIONS AND PERSPECTIVES ge
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190 BIBLIOGRAPHY [9] O. E. Barndorf
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192 BIBLIOGRAPHY [33] F. Delbaen, P
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194 BIBLIOGRAPHY [57] P. Jorion, On
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196 BIBLIOGRAPHY [81] S. Raible, L
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198 BIBLIOGRAPHY
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200 INDEX tightness, 40 levycalibra
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