Inverse Problem Stability of a Continuous-in-Time Financial Model
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2019
http://actams.wipm.ac.cn
INVERSE PROBLEM STABILITY OF A CONTINUOUS-IN-TIME FINANCIAL MODEL∗ Tarik CHAKKOUR Piaf INRA, Site de Crou¨el 5, Chemin de Beaulieu, Clermont-Ferrand 63000, France E-mail : [email protected] Abstract 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 ]). Key words
inverse problem; stability; mathematical model; Fredholm operator
2010 MR Subject Classification
1
45Q05; 60H25; 47B80
Introduction
In the last two decades, the theory and practice of inverse problems have been developed in many scientific domains. Consequently, it is rapidly growing, if not exploding. Moreover, document [1] showed how much researchers contributed to this field. Many inverse problems arising in scientific domains present numerical instability: the noise affecting the data may produce arbitrarily large errors in the solutions. In other words, these problems are ill-posed in the sense of Hadamard. The concept of ill-posedness was introduced by Hadamard [7] in the field of partial differential equations. We mention the book on the mathematics of ill-posed problems by Tikhonov and Arsenin [6]. We bluild in previous works [2, 5] the continuous-in-time model which is based on using the mathematical tools such convolution and integration. Indeed, this model uses measures over time interval to describe loan scheme, reimbursement scheme and interest payment scheme. The model contains some financial quantities. For instance, the repayment pattern measure γ˜ is a non-negative measure with total mass which equals 1, the algebraic spending measure σ ˜ is defined such that the difference between spendings and incomes required to satisfy the current needs between times t1 and t2 is Z t2 σ ˜, (1.1) t1
∗ Received
March 11, 2017; revised February 7, 2019.
1424
ACTA MATHEMATICA SCIENTIA
Vol.39 Ser.B
and the loan measure κ ˜ is defined such that the amount borrowed between times t1 and t2 is Z t2 κ ˜. (1.2) t1
When measure γ˜ is absolutely continuous with respect to the Lebesgue measure dt. This means that it read γ(t)dt, where t is the variable in R. The work [2] proposed the resolution of the inverse problem over the space of square-integrable functions when density γ is equal to Θ1γ over time period [0, Θγ ] and to 0 elsewhere. In papers [4, 13], we used a mathematical framework to discuss an inverse problem of determining the Loan Measure κ ˜ from algebraic spending measure σ ˜ . This inverse problem was used in [2] on simplified
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