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276 Y. Achdou ∂u ∂t ∈ L2 ((0,T)×R + ), such that u(t =0)=u ◦ and for all v ∈ K with bounded support in x, 〈〉 ∂u ∂t + Au + rx, v − u ≥ 0, for a.a. t>0. (57) There exists ˇX >0 such that u(t, x) =0, ∀t ∈ [0,T], x ≥ ˇX, (58) The function u is non-increasing with respect to x and non-decreasing with respect to t and there exists a non-decreasing continuous function γ :(0,T] → (S, ˇX), such that for all t ∈ (0,T), {x >0 s.t. u(t, x) =u ◦ (x)} =[γ(t), +∞). References [Ach05] Y. Achdou. An inverse problem for a parabolic variational inequality arising in volatility calibration with American options. SIAM J. Control Optim., 43(5):1583–1615 (electronic), 2005. [Ach06] Y. Achdou. An inverse problem for a parabolic variational inequality with an integro-differential operator arising in the calibration of Lévy processes with American options. Submitted, 2006. [Ada75] R. A. Adams. Sobolev spaces, volume 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich Publishers], New York, 1975. [AP05a] Y. Achdou and O. Pironneau. Computational methods for option pricing, volume 30 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. [AP05b] Y. Achdou and O. Pironneau. Numerical procedure for calibration of volatility with American options. Appl. Math. Finance, 12(3):201–241, 2005. [BL84] A. Bensoussan and J.-L. Lions. Impulse control and quasivariational inequalities. µ. Gauthier-Villars, Montrouge, 1984. Translated from the French by J. M. Cole. [BS73] F. Black and M. S. Scholes. The pricing of options and corporate liabilities,. Journal of Political Economy,, 81:637–654, 1973. [CT04] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. [CV03] R. Cont and E. Voltchkova. Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. Rapport Interne 513, CMAP, Ecole Polytechnique, 2003. [CV04] R. Cont and E. Voltchkova. Integro-differential equations for option prices in exponential Lévy models. Rapport Interne 547, CMAP, Ecole Polytechnique, 2004. [Dup97] B. Dupire. Pricing and hedging with smiles. In Mathematics of derivative securities (Cambridge, 1995), pages 103–111. Cambridge Univ. Press, Cambridge, 1997.
Calibration of Lévy Processes with American Options 277 [Hin01] [IK00] [JLL90] [Lio69] [Lio73] [LL97] [Mer73] [MNS03] [MP84] [MSW04] [MvPS04] [Pha98] [Pir84] M. Hintermüller. Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization. M2AN Math. Model. Numer. Anal., 35(1):129–152, 2001. K. Ito and K. Kunisch. Optimal control of elliptic variational inequalities. Appl. Math. Optim., 41(3):343–364, 2000. P. Jaillet, D. Lamberton, and B. Lapeyre. Variational inequalities and the pricing of American options. Acta Appl. Math., 21(3):263–289, 1990. J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969. J.-L. Lions. Perturbations singulières dans les problèmes aux limites et en contrôle optimal, volume 323 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, 1973. D. Lamberton and B. Lapeyre. Introduction au calcul stochastique appliqué à la finance. Ellipses, 1997. R. C. Merton. Theory of rational option pricing. Bell J. Econom. and Management Sci., 4:141–183, 1973. A.-M. Matache, P.-A. Nitsche, and C. Schwab. Wavelet Galerkin pricing of American options on Lévy driven assets. 2003. Research Report SAM 2003-06. F. Mignot and J.-P. Puel. Contrôle optimal d’un système gouverné par une inéquation variationnelle parabolique. C. R. Acad. Sci. Paris Sér. I Math., 298(12):277–280, 1984. A.-M. Matache, C. Schwab, and T. P. Wihler. Fast numerical solution of parabolic integro-differential equations with applications in finance. Technical report, IMA University of Minnesota, 2004. Reseach report No. 1954. A.-M. Matache, T. von Petersdoff, and C. Schwab. Fast deterministic pricing of Lévy driven assets. Mathematical Modelling and Numerical Analysis, 38(1):37–72, 2004. H. Pham. Optimal stopping of controlled jump-diffusion processes: A viscosity solution approach. Journal of Mathematical Systems, 8(1):1– 27, 1998. O. Pironneau. Optimal shape design for elliptic systems. Springer Series in Computational Physics. Springer-Verlag, New York, 1984.
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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96 I. Sazonov et al. To provide a p
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98 I. Sazonov et al. In the first s
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102 I. Sazonov et al. H z 1 exact F
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104 I. Sazonov et al. (a) (b) Fig.
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106 I. Sazonov et al. Scattering Wi
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108 I. Sazonov et al. 6.4 Scatterin
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110 I. Sazonov et al. (a) (b) Fig.
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112 I. Sazonov et al. [MHP96] K. Mo
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114 R. Sanders and A.M. Tesdall I R
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116 R. Sanders and A.M. Tesdall imp
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118 R. Sanders and A.M. Tesdall alo
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120 R. Sanders and A.M. Tesdall (a)
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122 R. Sanders and A.M. Tesdall D C
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124 R. Sanders and A.M. Tesdall 8.6
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126 R. Sanders and A.M. Tesdall 0.3
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128 R. Sanders and A.M. Tesdall [TR
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132 S. Lapin et al. Ω R γ Ω 2 Γ
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134 S. Lapin et al. ∫ ∂ 2 ∫
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136 S. Lapin et al. Ω R γ Fig. 3.
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138 S. Lapin et al. 4 Energy Inequa
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140 S. Lapin et al. 5 Numerical Exp
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142 S. Lapin et al. Fig. 6. Contour
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144 S. Lapin et al. Fig. 9. Obstacl
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Domain Decomposition and Electronic
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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188 A. Bonito et al. of the model a
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190 A. Bonito et al. Fig. 1. The sp
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192 A. Bonito et al. 3 1 16 4 1 1 1
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194 A. Bonito et al. ∫ v n+1 h
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196 A. Bonito et al. −pn +2µD(v)
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198 A. Bonito et al. each of its pa
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200 A. Bonito et al. The normal vec
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202 A. Bonito et al. with initial c
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204 A. Bonito et al. Fig. 8. Jet bu
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206 A. Bonito et al. References [AM
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208 A. Bonito et al. [Set96] J. A.
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210 J. Hao et al. due to shear flow
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212 J. Hao et al. The backward reac
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214 J. Hao et al. where u and p den
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216 J. Hao et al. * * * * * * * * *
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218 J. Hao et al. and solve for V n
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