In this work, we study the inverse problem stability of the continuous-in-time model which is designed to be used for the finances of public institutions. We discuss this study with determining the Loan measure from algebraic spending measure in Radon measure space M([tI, Θmax]), and in Hilbert space L2([tI, Θmax]) when they are density measures. For this inverse problem we prove the uniqueness theorem, obtain a procedure for constructing the solution and provide necessary and sufficient conditions for the solvability of the inverse problem in L2([tI, Θmax]).
Tarik CHAKKOUR
. INVERSE PROBLEM STABILITY OF A CONTINUOUS-IN-TIME FINANCIAL MODEL[J]. Acta mathematica scientia, Series B, 2019
, 39(5)
: 1423
-1439
.
DOI: 10.1007/s10473-019-0519-5
[1] Baltes H P. Progress in inverse optical problems//Inverse Scattering Problems in Optics, Topics in Current Physics. Springer 1980:1-13. https://doi.org/10.1007/978-3-642-81472-31
[2] Chakkour T, Frenod E. A continuous-in-time financial model. Mathematical Finance Letters, 2016, 2016:Article ID 2. http://scik.org/index.php/mfl/article/view/2510
[3] Chakkour T. Some notes about the continuous-in-time financial model. Abstract and Applied Analysis, 2017, 2017, https://doi.org/10.1155/2017/6985820
[4] Chakkour T, Frenod E. Inverse problem and concentration method of a continuous-in-time financial model. International Journal of Financial Engineering, 2016, 3(2):1650016. https://doi.org/10.1142/S242478631650016X
[5] Chakkour T. Implementing some mathematical operators for a continuous-in-time financial model. Engineering Mathematics Letters, 2017, 2017:Article ID 2. http://scik.org/index.php/eml/article/view/3060
[6] Andrey N T, Vasily Y A. Solutions of Ill-posed Problems. New York:Winston, 1977
[7] Hadamard J. Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, 1923
[8] la Sen M D. Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. Journal of Mathematical Analysis and Applications, 2006, 321(2):621-650
[9] Ceccki M, Furi M, Marini M. On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals. Nonlinear Analysis:Theory, Methods & Applications, 1985, 9(2):171-180
[10] Becker R I. Periodic solutions of semilinear equations of evolution of compact type. Journal of Mathematical Analysis and Applications, 1981, 82(1):33-48
[11] Freiling G, Yurko V. Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point. Inverse Problems, 2002, 18(3):757-773
[12] Sun Z Q, Uhlmann G. Recovery of singularities for formally determined inverse problems. Communications in Mathematical Physics, 1993, 153(3):431-445
[13] Chakkour T. Some inverse problem remarks of a continuous-in-time financial model in L1([tI, Θmax]). Preprint, 2019. https://hal.archives-ouvertes.fr/hal-01543421/
