Drift coefficient inversion problem of Kolmogorov-type equation

Liu Yang; Lijun Yin; Zuicha Deng; Liu Yang; Lijun Yin; Zuicha Deng

doi:10.3934/math.2021205

AIMS Mathematics

2021, Volume 6, Issue 4: 3432-3454. doi: 10.3934/math.2021205

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Research article

Drift coefficient inversion problem of Kolmogorov-type equation

1.
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
2.
Computer Science and Technology Experimental Teaching Center, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Received: 24 October 2020 Accepted: 08 January 2021 Published: 20 January 2021
MSC : 35R30, 49J20

Kolmogorov-type equations often appear in stochastic analysis and have important applications in financial derivatives pricing, stochastic control and other fields. In this paper, we consider an inverse problem of reconstructing drift coefficient in a Kolmogorov-type equation. Being different from other works, the unknown drift coefficient is related to both temporal and spatial variables, which makes theoretical analysis rather difficult. Until now, documents dealt with evolutional inverse drift problems are quite few. Inspired by the Rothe's idea, we introduce a new time semi-discrete scheme to find the optimal solution at each time layer. Then we construct an approximate solution of the unknown drift coefficient and strictly analyze its convergence. After establishing the necessary conditions for the limit minimizer, we prove the uniqueness and stability of the global optimal solution.
- Kolmogorov-type equation,
- inverse problem,
- drift coefficient,
- optimal control,
- uniqueness
Citation: Liu Yang, Lijun Yin, Zuicha Deng. Drift coefficient inversion problem of Kolmogorov-type equation[J]. AIMS Mathematics, 2021, 6(4): 3432-3454. doi: 10.3934/math.2021205

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Abstract

Kolmogorov-type equations often appear in stochastic analysis and have important applications in financial derivatives pricing, stochastic control and other fields. In this paper, we consider an inverse problem of reconstructing drift coefficient in a Kolmogorov-type equation. Being different from other works, the unknown drift coefficient is related to both temporal and spatial variables, which makes theoretical analysis rather difficult. Until now, documents dealt with evolutional inverse drift problems are quite few. Inspired by the Rothe's idea, we introduce a new time semi-discrete scheme to find the optimal solution at each time layer. Then we construct an approximate solution of the unknown drift coefficient and strictly analyze its convergence. After establishing the necessary conditions for the limit minimizer, we prove the uniqueness and stability of the global optimal solution.

References

[1]	R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
[2]	I. Bushuyev, Global uniqueness for inverse problems with final observation, Inverse Probl., 11 (1995), L11–L16. doi: 10.1088/0266-5611/11/4/001
[3]	J. R. Cannon, Y. Lin, S. Xu, Numerical procedure for the determination of an unknown coefficient in semilinear parabolic partial differential equations, Inverse Probl., 10 (1994), 227–243. doi: 10.1088/0266-5611/10/2/004
[4]	J. R. Cannon, Y. Lin, An inverse problem of finding a parameter in a semilinear heat equation, J. Math. Anal. Appl., 145 (1990), 470–484. doi: 10.1016/0022-247X(90)90414-B
[5]	Q. Chen, J. J. Liu, Solving an inverse parabolic problem by optimization from final measurement data, J. Comput. Appl. Math., 193 (2006), 183–203. doi: 10.1016/j.cam.2005.06.003
[6]	M. Choulli, M. Yamamoto, Generic well-posedness of an inverse parabolic problem - the Hölder space approach, Inverse Probl., 12 (1996), 195–205. doi: 10.1088/0266-5611/12/3/002
[7]	M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. Diff. Equ., 21 (2005), 611–622. doi: 10.1002/num.20055
[8]	M. Dehghan, Determination of a control function in three-dimensional parabolic equations, Math. Comput. Simul., 61 (2003), 89–100. doi: 10.1016/S0378-4754(01)00434-7
[9]	Z. C. Deng, L. Yang, J. N. Yu, G. W. Luo, Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method, J. Math. Anal. Appl., 362 (2010), 210–223. doi: 10.1016/j.jmaa.2009.08.042
[10]	Z. C. Deng, J. N. Yu, L. Yang, Optimization method for an evolutional type inverse heat conduction problem, J. Phys. A: Math. Theor., 41 (2008), 035201. doi: 10.1088/1751-8113/41/3/035201
[11]	Z. C. Deng, J. N. Yu, L. Yang, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simul., 77 (2008), 421–435.
[12]	H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Dordrecht: Kluwer Academic Publishers, 1996.
[13]	L. C. Evans, Partial Differential Equations, American Math Society, 1998.
[14]	A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ: Prentice-Hall, 1964.
[15]	G. L. Gong, Stochastic differential equation and its application, Tsinghua University Press, Beijing, 2008.
[16]	V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998.
[17]	V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185–209. doi: 10.1002/cpa.3160440203
[18]	L. S. Jiang, Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inverse Probl., 17 (2001), 137–155. doi: 10.1088/0266-5611/17/1/311
[19]	L. S. Jiang, Q. H. Chen, L. J. Wang, J. E. Zhang, A new well-posed algorithm to recover implied local volatility, Quant. Finance, 3 (2003), 451–457. doi: 10.1088/1469-7688/3/6/304
[20]	L. S. Jiang, B. J. Bian, Identifying the principal coefficient of parabolic equations with non-divergent form, J. Phys.: Conf. Ser., 12 (2005), 58–65. doi: 10.1088/1742-6596/12/1/006
[21]	L. S. Jiang, C. L. Xu, X. M. Ren, S. H. Li, Mathematical model and case analysis of financial derivatives pricing, Higher Education Press, Beijing, 2008.
[22]	B. Kaltenbacher, M. V. Klibanov, An inverse problem for a nonlinear parabolic equation with applications in population dynamics and magnetics, SIAM J. Math. Anal., 39 (2008), 1863–1889. doi: 10.1137/070685907
[23]	Y. L. Keung, J. Zou, Numerical identifications of parameters in parabolic systems, Inverse Probl., 14 (1998), 83–100. doi: 10.1088/0266-5611/14/1/009
[24]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problem, Springer, New York, 1999.
[25]	O. Ladyzenskaya, V. Solonnikov, N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type, Providence, KI: American Mathematical Society, 1968.
[26]	W. Rundell, The determination of a parabolic equation from initial and final data, Proc. Am. Math. Soc., 99 (1987), 637–642. doi: 10.1090/S0002-9939-1987-0877031-4
[27]	A. A. Samarskii, P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Berlin, Germany, 2007.
[28]	A. Tikhonov, V. Arsenin, Solutions of Ill-posed Problems, Geology Press, Beijing, 1979.
[29]	D. Trucu, D. B. Ingham, D. Lesnic, Inverse temperature-dependent perfusion coefficient reconstruction, Int. J. Non-Linear Mech., 45 (2010), 542–549. doi: 10.1016/j.ijnonlinmec.2010.02.004
[30]	D. Trucu, D. B. Ingham, D. Lesnic, Reconstruction of the space- and time-dependent blood perfusion coefficient in bio-heat transfer, Heat Transfer Eng., 32 (2011), 800–810. doi: 10.1080/01457632.2011.525430
[31]	D. Trucu, D. B. Ingham, D. Lesnic, Space-dependent perfusion coefficient identification in the transient bio-heat equation, J. Eng. Math., 67 (2010), 307–315. doi: 10.1007/s10665-009-9319-6
[32]	Z. Q. Wu, J. X. Yin, C. P. Wang, Introduction to elliptic and parabolic equations, Science Press, Beijing, 2003.
[33]	M. Yamamoto, J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Probl., 17 (2001), 1181–1202. doi: 10.1088/0266-5611/17/4/340
[34]	L. Yang, J. N. Yu, Z. C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Appl. Math. Modell., 32 (2008), 1984–1995. doi: 10.1016/j.apm.2007.06.025

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