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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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

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