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On Inverse Problem of Determination of the Coefficient in the Black-Scholes Type Equation

  • Vitaly L. KamyninEmail author
  • Tatiana I. Bukharova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

We prove the existence and uniqueness theorems for inverse problem of determination of the lower coefficient in the Black-Scholes type equation with additional condition of integral observation. These results are based on the investigation of unique solvability of corresponding direct problem which is of independent interest. We give the example of the inverse problem for which the conditions of the theorems proved are fulfilled.

Notes

Acknowledgements

This work was partially supported by the Program of competitiveness increase of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); contract No. 02.a03.21.0005, 27.08.2013.

References

  1. 1.
    Kruzhkov, S.N.: Quasilinear parabolic equations and systems with two independent variables. Trudy Sem. im. I.G. Petrovskogo 5, 217–272 (1979)MathSciNetGoogle Scholar
  2. 2.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hull, J.: Options, Futures and Other Derivatives. Prentice Hall, Upper Saddle River (2005)zbMATHGoogle Scholar
  4. 4.
    Fichera, G.: Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine. Atti Accad. Nazionale dei Lincei. Mem. Cl. Sci. Fis. Mat. Natur. Ser. I(8) 5, 1–30 (1956)zbMATHGoogle Scholar
  5. 5.
    Oleǐnik, O.A., Radkevič, E.A.: Second Order Differential Equations with Nonnegative Characteristic Form. AMS, Rhode Island and Plenum Press, New York (1973)CrossRefGoogle Scholar
  6. 6.
    Deng, Z.C., Yang, L.: An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation. J. Comput. Appl. Math. 235, 4404–4417 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deng, Z.C., Yang, L.: An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation. Chin. Ann. Math. Ser. B. 35B(3), 355–382 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bouchouev, I., Isakov, V.: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Prob. 15(3), 95–116 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lishang, J., Yourshan, T.: Identifying the volatibility of underlying assets from option prices. Inverse Prob. 17(1), 137–155 (2001)CrossRefGoogle Scholar
  10. 10.
    Lishang, J., Qihong, C., Lijun, W., Zhang, J.E.: A new well-posed algorithm to recover implied local volatibility. Quant. Financ. 3(6), 451–457 (2003)CrossRefGoogle Scholar
  11. 11.
    Prilepko, A.I., Kamynin, V.L., Kostin, A.B.: Inverse source problem for parabolic equation with the condition of integral observation in time. J. Inverse III-posed Prob. 26(4), 523–539 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bukharova, T.I., Kamynin, V.L.: Inverse problem of determining the absorption coefficient in the multidimensional heat equation with unlimited minor coefficients. Comput. Math. Math. Phys. 55(7), 1183–1195 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lyusternik, L.A., Sobolev, V.I.: Kratkii Kurs Functcional’nogo Analiza (Brief Course of Functional Analysis). Vysshaya Shkola, Moscow (1982)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia

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