Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Vinicius Albani

    * Corresponding author: Vinicius Albani 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 65J22, 47J06; Secondary: 35R30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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 CezaroO. 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. EnglK. 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=
    [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. JacksonE. 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.
  • 加载中

Article Metrics

HTML views(471) PDF downloads(533) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint