American Institute of Mathematical Sciences

Advanced
February  2019, 13(1): 211-229. doi: 10.3934/ipi.2019012

A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles

Vinicius Albani 1,, and  Adriano De Cezaro 2,

1. 

Dept of Mathematics, Federal University of Santa Catarina, Campus Trindade, Florianopolis, SC 88.040-900, Brazil
2. 

IMEF, Federal University of Rio Grande, Av. Italia km 8, Rio Grande, RS 96201-900, Brazil

* Corresponding author: Vinicius Albani

Received  June 2018 Revised  September 2018 Published  December 2018

We state sufficient conditions for the uniqueness of minimizers of Tikhonov-type functionals. We further explore a connection between such results and the well-posedness of Morozov-like discrepancy principle. Moreover, we find appropriate conditions to apply such results to the local volatility surface calibration problem.

Keywords: Tikhonov-type regularization, uniqueness of minimizers, discrepancy principles, local volatility calibration.
Mathematics Subject Classification: Primary: 65J22, 47J06; Secondary: 35R30.
Citation: Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems & Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012
References:
[1]

Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, Frontiers in Applied Mathematics, SIAM, 2005. doi: 10.1137/1.9780898717495.  Google Scholar

[2]

R. A. Adams and J. J. F. Founier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[3]

V. Albani, A. De Cezaro and J. Zubelli, On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy, Inverse Problems and Imaging, 10 (2016), 1–25, URL https://aimsciences.org/journals/displayArticlesnew.jsp?paperID=12262. doi: 10.3934/ipi.2016.10.1.  Google Scholar

[4]

V. Albani, A. De Cezaro and J. Zubelli, Convex Regularization of Local Volatility Estimation, Int. J. Theor. Appl. Finance, 20 (2017), 1750006, 37pp. doi: 10.1142/S0219024917500066.  Google Scholar

[5]

S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp. doi: 10.1088/0266-5611/26/2/025001.  Google Scholar

[6]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007, 18pp. doi: 10.1088/0266-5611/27/10/105007.  Google Scholar

[7]

M. Avellaneda, Minimum-Relative-Entropy Calibration of Asset-Pricing Models, Int. J. Theor. Appl. Finance, 1 (1998), 447-472.   Google Scholar

[8]

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[9]

I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17.  doi: 10.1088/0266-5611/13/5/001.  Google Scholar

[10]

I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116.  doi: 10.1088/0266-5611/15/3/201.  Google Scholar

[11]

R. Carmona and S. Nadtochyi, Local volatility dynamic models, Finance Stoch., 13 (2009), 1-48.  doi: 10.1007/s00780-008-0078-4.  Google Scholar

[12]

G. Chavent and K. Kunisch, On Weakly Nonlinear Inverse Problems, SIAM J. Appl. Math, 56 (1996), 542-572.  doi: 10.1137/S0036139994267444.  Google Scholar

[13]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.  Google Scholar

[14]

S. Crepey, Calibration of the Local Volatility in a Generalized Black-Scholes Model Using Tikhonov Regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206.  doi: 10.1137/S0036141001400202.  Google Scholar

[15]

A. De Cezaro, O. Scherzer and J. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates, Nonlinear Anal., 75 (2012), 2398-2415.  doi: 10.1016/j.na.2011.10.037.  Google Scholar

[16]

A. De Cezaro and J. P. Zubelli, The tangential cone condition for the iterative calibration of local volatility surfaces, IMA J. Appl. Math., 80 (2013), 212-232.  doi: 10.1093/imamat/hxt037.  Google Scholar

[17]

E. Derman and I. Kani, Riding on a Smile, Risk, 7 (1994), 32-39.   Google Scholar

[18]

B. Dupire, Pricing with a smile, Risk Magazine, 7 (1994), 18-20.   Google Scholar

[19]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.  doi: 10.1088/0266-5611/21/3/014.  Google Scholar

[20]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[21]

H. Engl and W. Grever, Using the L-curve for determining optimal regularization parameters, Numer. Math., 69 (1994), 25-31.  doi: 10.1007/s002110050078.  Google Scholar

[22]

H. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Paraboloc Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[24]

P. C. Hansen, Analysis of Discrete Ill-Posed Problems by Means of the L-Curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[25]

P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in Computational inverse problems in electrocardiology, WIT Press, 2000, URL http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.499.7234&rep=rep1&type=pdf. Google Scholar

[26]

B. Hofmann and R. Krämer, On maximum entropy regularization for a specific inverse problem of option pricing, J. Inverse Ill-Posed Probl., 13 (2005), 41-63.  doi: 10.1515/1569394053583739.  Google Scholar

[27]

N. Jackson, E. Süli and S. Howison, Computation of Deterministic Volatility Surfaces, J. Comput. Finance, 2 (1998), 5-32.   Google Scholar

[28]

F. Margotti and A. Rieder, An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method, J. Inverse Ill-Posed Probl., 23 (2014), 373-392.  doi: 10.1515/jiip-2014-0035.  Google Scholar

[29]

V. Morozov, On the solution of functional equations by the method of regularization, Dokl. Math., 7 (1966), 414-417.   Google Scholar

[30]

J. Qi-Niam, Applications of the Modified Discrepancy Principle to Tikhonov Regularization of Nonlinear Ill-Posed Problems, SIAM J. Numer. Anal., 36 (1999), 475-490.  doi: 10.1137/S0036142997315470.  Google Scholar

[31]

O. Scherzer, The use of Morozov discrepancy principle for Tikhonov regularization for solving non-linear ill-posed problems, Computing, 51 (1993), 45-60.  doi: 10.1007/BF02243828.  Google Scholar

[32]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, vol. 167 of Applied Mathematical Sciences, Springer, New York, 2009.  Google Scholar

[33]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Walter de Gruyter, 2012. doi: 10.1515/9783110255720.  Google Scholar

show all references

References:
[1]

Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, Frontiers in Applied Mathematics, SIAM, 2005. doi: 10.1137/1.9780898717495.  Google Scholar

[2]

R. A. Adams and J. J. F. Founier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[3]

V. Albani, A. De Cezaro and J. Zubelli, On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy, Inverse Problems and Imaging, 10 (2016), 1–25, URL https://aimsciences.org/journals/displayArticlesnew.jsp?paperID=12262. doi: 10.3934/ipi.2016.10.1.  Google Scholar

[4]

V. Albani, A. De Cezaro and J. Zubelli, Convex Regularization of Local Volatility Estimation, Int. J. Theor. Appl. Finance, 20 (2017), 1750006, 37pp. doi: 10.1142/S0219024917500066.  Google Scholar

[5]

S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp. doi: 10.1088/0266-5611/26/2/025001.  Google Scholar

[6]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007, 18pp. doi: 10.1088/0266-5611/27/10/105007.  Google Scholar

[7]

M. Avellaneda, Minimum-Relative-Entropy Calibration of Asset-Pricing Models, Int. J. Theor. Appl. Finance, 1 (1998), 447-472.   Google Scholar

[8]

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[9]

I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17.  doi: 10.1088/0266-5611/13/5/001.  Google Scholar

[10]

I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116.  doi: 10.1088/0266-5611/15/3/201.  Google Scholar

[11]

R. Carmona and S. Nadtochyi, Local volatility dynamic models, Finance Stoch., 13 (2009), 1-48.  doi: 10.1007/s00780-008-0078-4.  Google Scholar

[12]

G. Chavent and K. Kunisch, On Weakly Nonlinear Inverse Problems, SIAM J. Appl. Math, 56 (1996), 542-572.  doi: 10.1137/S0036139994267444.  Google Scholar

[13]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.  Google Scholar

[14]

S. Crepey, Calibration of the Local Volatility in a Generalized Black-Scholes Model Using Tikhonov Regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206.  doi: 10.1137/S0036141001400202.  Google Scholar

[15]

A. De Cezaro, O. Scherzer and J. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates, Nonlinear Anal., 75 (2012), 2398-2415.  doi: 10.1016/j.na.2011.10.037.  Google Scholar

[16]

A. De Cezaro and J. P. Zubelli, The tangential cone condition for the iterative calibration of local volatility surfaces, IMA J. Appl. Math., 80 (2013), 212-232.  doi: 10.1093/imamat/hxt037.  Google Scholar

[17]

E. Derman and I. Kani, Riding on a Smile, Risk, 7 (1994), 32-39.   Google Scholar

[18]

B. Dupire, Pricing with a smile, Risk Magazine, 7 (1994), 18-20.   Google Scholar

[19]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.  doi: 10.1088/0266-5611/21/3/014.  Google Scholar

[20]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[21]

H. Engl and W. Grever, Using the L-curve for determining optimal regularization parameters, Numer. Math., 69 (1994), 25-31.  doi: 10.1007/s002110050078.  Google Scholar

[22]

H. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Paraboloc Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[24]

P. C. Hansen, Analysis of Discrete Ill-Posed Problems by Means of the L-Curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[25]

P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in Computational inverse problems in electrocardiology, WIT Press, 2000, URL http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.499.7234&rep=rep1&type=pdf. Google Scholar

[26]

B. Hofmann and R. Krämer, On maximum entropy regularization for a specific inverse problem of option pricing, J. Inverse Ill-Posed Probl., 13 (2005), 41-63.  doi: 10.1515/1569394053583739.  Google Scholar

[27]

N. Jackson, E. Süli and S. Howison, Computation of Deterministic Volatility Surfaces, J. Comput. Finance, 2 (1998), 5-32.   Google Scholar

[28]

F. Margotti and A. Rieder, An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method, J. Inverse Ill-Posed Probl., 23 (2014), 373-392.  doi: 10.1515/jiip-2014-0035.  Google Scholar

[29]

V. Morozov, On the solution of functional equations by the method of regularization, Dokl. Math., 7 (1966), 414-417.   Google Scholar

[30]

J. Qi-Niam, Applications of the Modified Discrepancy Principle to Tikhonov Regularization of Nonlinear Ill-Posed Problems, SIAM J. Numer. Anal., 36 (1999), 475-490.  doi: 10.1137/S0036142997315470.  Google Scholar

[31]

O. Scherzer, The use of Morozov discrepancy principle for Tikhonov regularization for solving non-linear ill-posed problems, Computing, 51 (1993), 45-60.  doi: 10.1007/BF02243828.  Google Scholar

[32]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, vol. 167 of Applied Mathematical Sciences, Springer, New York, 2009.  Google Scholar

[33]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Walter de Gruyter, 2012. doi: 10.1515/9783110255720.  Google Scholar

[1]

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems & Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1
[2]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[3]

Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1699-1729. doi: 10.3934/jimo.2019025
[4]

Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems & Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291
[5]

Abhishake Rastogi. Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4111-4126. doi: 10.3934/cpaa.2020183
[6]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036
[7]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102
[8]

Vinicius Albani, Uri M. Ascher, Xu Yang, Jorge P. Zubelli. Data driven recovery of local volatility surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 799-823. doi: 10.3934/ipi.2017038
[9]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
[10]

Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011
[11]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013
[12]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[13]

Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008
[14]

Lishang Jiang, Baojun Bian. The regularized implied local volatility equations -A new model to recover the volatility of underlying asset from observed market option price. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2017-2046. doi: 10.3934/dcdsb.2012.17.2017
[15]

Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control & Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026
[16]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[17]

Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 147-165. doi: 10.3934/dcdss.2013.6.147
[18]

Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 251-265. doi: 10.3934/dcdss.2019018
[19]

Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems & Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107
[20]

Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (130)
  • HTML views (197)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences