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Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles
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 |
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.
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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