# On Inverse Problems for Strongly Degenerate Parabolic Equations under the Integral Observation Condition

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### Abstract

Existence and uniqueness theorems for inverse problems of determining the right-hand side and lowest coefficient in a degenerate parabolic equation with two independent variables are proved. It is assumed that the leading coefficient of the equation degenerates at the side boundary of the domain and the order of degeneracy with respect to the variable \(x\) is not lower than 2. Thus, the Black–Scholes equation, well-known in financial mathematics, is admitted. These results are based on the study of the unique solvability of the corresponding direct problem, which is also of independent interest.

## Keywords:

direct and inverse problems integral observation condition degenerate parabolic equations## Notes

### ACKNOWLEDGMENTS

This work was carries out within the Program for Increasing the Competitiveness of the National Research Nuclear University MEPhI, project no. 02.A03.21.0005 of August 27, 2013.

I am grateful to L. Vulkov and A.B. Kostin for useful discussion.

## REFERENCES

- 1.F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Political Econ.
**81**, 637–659 (1973).MathSciNetCrossRefzbMATHGoogle Scholar - 2.J. Hull,
*Options, Futures, and Other Derivations*(Prentice Hall, Upper Saddle River, N.J., 2005).Google Scholar - 3.P. Wilmott, J. Dewynne, and S. Howison,
*Option Pricing: Mathematical Models and Computation*(Oxford Financial, Oxford, 1993).zbMATHGoogle Scholar - 4.I. Bouchouev and V. Isakov, “Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets,” Inverse Probl.
**15**(3), R95–R116 (1999).MathSciNetCrossRefzbMATHGoogle Scholar - 5.L. Jiang and Y. Tao, “Identifying the volatility of underlying assets from option prices,” Inverse Probl.
**17**(1), 137–155 (2001).MathSciNetCrossRefzbMATHGoogle Scholar - 6.L. Jiang, Q. Chen, L. Wang, and J. E. Zhang, “A new well-posed algorithm to recover implied local volatility,” Quant. Finance
**3**(6), 451–457 (2003).MathSciNetCrossRefzbMATHGoogle Scholar - 7.P. Cannarsa, P. Martinez, and J. Vancostenoble,
*Global Carleman Estimates for Degenerate Parabolic Operators with Applications*(Am. Math. Soc., Providence, R.I., 2015).zbMATHGoogle Scholar - 8.G. Fichera, “Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine,” Atti Accad. Naz. Lincei. Memorie Classe Sci. Fis. Mat. Natur. Ser. I(8)
**5**(1), 1–30 (1956).Google Scholar - 9.O. A. Oleinik and E. V. Radkevich,
*Equations with a Nonnegative Characteristic Form*(Mosk. Gos. Univ., Moscow, 2010) [in Russian].Google Scholar - 10.Z. C. Deng and L. Yang, “An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation,” J. Comp. Appl. Math.
**235**, 4404–4417 (2011).Google Scholar - 11.Z. C. Deng and L. Yang, “An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation,” J. Chinese Ann. Math. Ser. B
**35**(3), 355–382 (2014).Google Scholar - 12.M. Ivanchov and N. Saldina, “An inverse problem for strongly degenerate heat equation,” J. Inverse Ill-posed Probl.
**14**(5), 465–480 (2006).MathSciNetCrossRefzbMATHGoogle Scholar - 13.P. Cannarsa, J. Tort, and M. Yamamoto, “Determination of source terms in degenerate parabolic equation,” Inverse Probl.
**26**(10), 105003 (2010).MathSciNetCrossRefzbMATHGoogle Scholar - 14.Z. C. Deng, K. Qian, X. B. Rao, and L. Yang, “An inverse problem of identifying the source coefficient in degenerate heat equation,” Inverse Probl. Sci. Eng.
**23**(3), 498–517 (2014).MathSciNetCrossRefzbMATHGoogle Scholar - 15.N. Huzyk, “Inverse Problem of determining the coefficients in degenerate parabolic equation,” Electron. J. Differ. Equations
**172**, 1–11 (2014).MathSciNetzbMATHGoogle Scholar - 16.V. L. Kamynin, “On the solvability of the inverse problem for determining the right-hand side of a degenerate parabolic equation with integral observation,” Math. Notes
**98**(5), 765–777 (2015).MathSciNetCrossRefzbMATHGoogle Scholar - 17.V. L. Kamynin, “Inverse problem of determining the right-hand side in a degenerating parabolic equation with unbounded coefficients,” Comput. Math. Math. Phys.
**57**(5), 833–842 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 18.S. N. Kruzhkov, “Quasilinear parabolic equations and systems with two independent variables,” Trudy Semin. im. I.G. Petrovskogo, No.
**5**, 217–272 (1979).Google Scholar - 19.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,
*Linear and Quasilinear Equations of Parabolic Type*(Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).Google Scholar - 20.O. A. Oleinik,
*Lectures on Partial Differential Equations*(Mosk. Gos. Univ., Moscow, 1976), Part 1 [in Russian].Google Scholar - 21.L. A. Lusternik and V. I. Sobolev,
*Elements of Functional Analysis*(Gordon and Breach, New York, 1968; Vysshaya Shkola, Moscow, 1982).Google Scholar