Some Notes about the Continuous-in-Time Financial Model

and Applied Analysis 3 Next, we simplify the third one as follows: ∫ tI (∫ tI κE (y) dy) κF (x) dx = ∫ tI (∫ tI κE (y) 1{y≤x}dy) κF (x) dx = ∫ tI κE (y) (∫ y κF (x) dx)dy. (10) The last one is simplified as follows: ∫ tI (∫ tI κE ⋆ γ (y) dy) κF (x) dx = ∫ tI (∫ tI (∫ tI γ (y − t) κE (t) dt) dy) κF (x) dx = ∫ tI (∫ tI 1{y≤x} (∫ tI γ (y − t) κE (t) 1{t≤y}dt) dy) ⋅ κF (x) dx = ∫ tI (∫ tI κE (t) 1{t≤x} (∫ tI γ (y − t) 1{t≤y≤x}dy)dt) ⋅ κF (x) dx = ∫ tI (∫ tI κF (x) 1{t≤x} (∫ tI γ (y − t) 1{t≤y≤x}dy)dx) ⋅ κE (t) dt. (11) According to relations (9), (10), and (11), expression (8) of inner product ⟨L[κE], κF⟩ reads ⟨L [κE] , κF⟩ = ∫ tI κE (y) (κF (y) − ∫ y γ (x − y) κF (x) dx − α∫ y κF (x) dx + α∫ y κF (x) (∫ y γ (z − y) dz) dx)dy. (12) Since operatorL is one such that ⟨L [κE] , κF⟩ = ⟨κE,L∗ [κF]⟩ , ∀κE ∈ L ([tI, Θmax − Θγ]) , ∀κF ∈ L ([tI, Θmax]) , (13) and, according to relation (12), operatorL is defined by L ∗ [κF] (y) = κF (y) − ∫ y γ (x − y) κF (x) dx − α∫ y κF (x) dx + α∫ y κF (x) ⋅ (∫ y γ (z − y) dz)dx = κF (y) − ∫ y κF (x) ⋅ (γ (x − y) + α − α∫ y γ (z − y) dz)dx.


Introduction
Linear inverse problems arise whenever throughout engineering and the mathematical sciences.In most applications, these problems are ill-conditioned or underdetermined.Consequently, over the last two decades, the theory and practice of inverse problems is rapidly growing, if not exploding in many scientific domains.The fundamental reason is that solutions to inverse problems describe important properties of solution in this theory and the development of sophisticated numerical techniques for its treating on a level of high complexity.We mention the paper [1] introduced by Hadamard in the field of ill-posed problems.
We built in previous work [2,3] the continuous-intime model which is designed to be used for the finances of public institutions.This model uses measures over time interval to describe loan scheme, reimbursement scheme, and interest payment scheme.Algebraic Spending Measure σ and Loan Measure κ are financial variables involved in the model.Measure σ is defined such that the difference between spending and incomes required to satisfy the current needs.Assume that measures σ and κ are absolutely continuous with respect to the Lebesgue measure .This means that they read () and   (), where  is the variable in R. We call  and   time densities.
Let  and  be normed spaces.Throughout this paper, L :  →  is a continuous linear application (in short, an operator).We say that the following problem: is well-posed if L is invertible and its inverse L −1 :  →  is continuous.In other words, the problem is said to be wellposed if the solution   depends continuously on .
Existence and uniqueness of a solution for all  ∈  (condition (2)) are equivalent to surjectivity and injectivity of L, respectively.Stability of the solution (condition (3)) amounts to continuity of L −1 .Conditions (2) and ( 3) are referred to as the Hadamard conditions.A problem which is not well-posed is said to be ill-posed.Operator L links between Algebraic Spending Density  and Loan Density   .If this operator is not invertible, solutions of the posed inversion problem can be brought.
In the recent papers [2,4], we study the inverse problem stability of the continuous-in-time model.We discuss this study with determining Loan Measure κ from Algebraic Spending Measure σ in Radon measure space, that is,  = M([ I , Θ max − Θ  ]) and  = M([ I , Θ max ]), and in Hilbert space, that is,  = L 2 ([ I , Θ max −Θ  ]) and  = L 2 ([ I , Θ max ]), when they are density measures.For this inverse problem we prove the uniqueness theorem in [4]; we obtain a procedure for constructing the solution and provide necessary and sufficient conditions for the solvability of the inverse problem.
We are motivated by a recently developed nonlinear inverse scale in Schwartz space.We refer the reader to [5,6], for applications of fast inversion formulas to inverse problems.Bauer and Lukas investigate in [5] some different frameworks for regularization of linear inverse problems when error is expected to be decreased at infinity.In the paper [6], Hansen investigates the approximation properties of regularized solutions to discrete ill-posed.They average decay to zero faster than the generalized singular values.
We show in this paper some results of this inverse problem in Schwartz space.We sketch the theoretical results that justify the mathematical well-posedness under some assumptions.The main result of this paper is to study the existence and uniqueness of solutions.We give an overview of properties for operator L, describing the computation of its image.
The rest of this paper is arranged as follows.In Section 2 we introduce the definition of operator L and others, and the mathematical properties of these operators are shown.We treat in Section 3 the spectrum of some operators involved in the model by determining the inverse of operator under some hypothesis.It is followed by enrichment of the model of variable rate in Section 4. In Section 5, we examine the concept of ill-posedness in Schwartz space in order to obtain interesting and useful solutions.

Properties of Operators
This section is devoted to explore mathematical properties of some operators involved in the model.Those properties will be useful for some aspects of the model implementation to come in the following.These operators are acting on measures over R. For that, we will also consider that M([ I , Θ max ]) is the set of Radon Measures over R, supported in [ I , Θ max ].In the sequel, we consider the case when all measures are density measures.The purpose is to compute the adjoint of these operators.We will be able to use some specific mathematical tools as inner product.We proceed by denoting L 2 ([ I , Θ max ]) the space of square-integrable functions over R having their support in [ I , Θ max ].We state the Repayment Pattern Density  as follows: where Θ  is a positive number such that We recall that we have shown the balanced equation given by equality (14) in [2].This equality consists in writing Loan Density   as a sum of Algebraic Spending Density  and densities associated with quantities that have to be repaid or paid.This equality yields with convolution equality defined by ( 9) in [2] to express density : From this, linear term of density  is defined by linear operator L acting on Loan Density given by The aim here is to compute operator L * .For that, we will compute inner product Abstract and Applied Analysis 3 Next, we simplify the third one as follows: The last one is simplified as follows: ⋅   () .(11) According to relations (9), (10), and (11), expression (8) of inner product ⟨L[  ],   ⟩ reads and, according to relation (12), operator L * is defined by The integration by parts states that inner product ⟨D[ I K ],   ⟩ is computed for any densities  I K and   in L 2 ([ I , Θ max ]) as follows: From this, operator D * is defined by Operator L we set out in relation ( 7) considered a constant rate.Nevertheless, if we consider in it a function  that depends on , the model becomes a financial model with variable rate.The only modification to make is to enrich (7) and ( 15) by writing Once this enrichment is done, using a variable rate makes operator D * expressed in terms of densities  and : By definition () is the rate at time .Operator D * given by relation ( 17) can be rewritten adding this enrichment Lemma 1.The image of operator L is such that Proof.In order to show equality (22), we will show that the kernel of operator L * is reduced to null set because of the following property: According to (14), if density   is in Ker(L * ), then, we get the following equation: Deriving, we get the ODE that the solution   is expressed as follows: The general solution to (25) is given by where initial condition   ( I ) stands for the value of density   at initial time  I .On the other hand, (24) is equivalent to Conversely, we will show that density   is zero.In the first place, replacing density   given by (26) in first term  1 of (27), we get Secondly, the second term  2 is a constant function due to its derivative which equals zero: (29) Consequently, the second term  2 is equal to a real constant  to be determined: The initial condition is obtained from integral equation defined by (30) with replacing  by Θ max , which implies that constant  is zero.It is concluded that Then, relations (27), (28), and (31) yield the following equality: If loan rate  is zero, then, according to (24), we obtain following integral equation: where expression of density   is determined from equality (26) as If density   given by ( 35) is coupled with (34), then,   ( I ) is zero allowing zero density   .We showed that density   is zero in both cases.From this, we can deduce that (33) is true, proving the lemma.

Spectrum of Operators
It is well known that the integral operators [7,8] possess a very rich structure theory, such that these operators played an important role in the study of operators on Hilbert Spaces.
The paper [9] and book [10] by Gil' deal with the spectra of a class of linear non-self-adjoint operators containing the Volterra operators.Since this operator is involved in the model, we use it in order to study the spectrum of some operators.It is shown in [11] that the spectrum of Volterra composition operator is consisting of zero only.This section is devoted to explore the spectrum of some operators involving the spectrum of Volterra.Defining linear operator Ṽ : ) by operator that is acting on Loan Density   , which is decomposed as a sum of operators L and Ṽ given by relations ( 7) and (36), respectively: Theorem 2. If density  has upper bound  − 2|| over its support: sup where  is a positive real satisfying then, operator L is invertible, where its inverse L −1 is given by where  is an operator defined by ) and by using definition (42) of operator , we get From this, we get which implies that Relation (45) shows that equality (41) is consistent due to the inverse of operator L being in L 2 ([ I , Θ max − Θ  ]).Next, we can show that operator Ṽ ∘  can be written in the form where kernel  is defined as Following equality yields with Fubini-Tonelli theorem to obtain (49) Abstract and Applied Analysis Consequently, operator ( Ṽ∘) (2) is written in following form: where We can verify by induction for  ≥ 2 that the recurrence expresses each operator ( Ṽ ∘ ) () as an integral operator which is written in following form: where kernel   is given as Now, we will show that  is a maximum of kernel  over By integrating each term of (60) over interval [ I , Θ max ], we get Consequently, Inequality (61) gives Since the series ∑ ≥1   of defined terms is convergent, then quantity ∑ ≥0 ( Ṽ ∘ ) () converges absolutely in L 2 ([ I , Θ max ]).Consequently, ∑ ≥0 ( Ṽ ∘ ) () exists and is finite in L 2 ([ I , Θ max ]).We recall that we have Since we have √ ≤ ( − 1)! for all integer  ≥ 2, inequality (61) implies that Coupling inequality (64) with the fact that ‖( Ṽ ∘ ) () ‖ L 2 ([ I ,Θ max ]) converges to 0 due to (40), we get Composing operators defined by (38) with operator , we get According to (66) and (67), we get That is, achieving equality (41) of the lemma.

Extensions in the Model of Variable Rate
We built in [2] the financial models that are used on simplified problems in order to show how they can be used in reality.This section is devoted to enrich the model in order to account for this reality.In particular, we will express the Algebraic Spending Density  I in the model with variable rate.The mathematical consistency of this density  I is analyzed.
The Current Debt Field K RD is related to Loan Measure κ and Repayment Measure ρK by the following Ordinary Differential Equation: The solution of this ODE is expressed as follows: Since the Interest Payment Density  I is related to the Current Debt Field by a proportionality relation: the Interest Payment Density  I can be expressed in terms of Loan Density   : Since Density  reads where operators L and D are defined in relations ( 18) and ( 19), respectively, we get equality (6).From this, we get the following Ordinary Differential Equation: Defining  RD (, ) the Current Debt at time  which is associated with the amount borrowed at time . RD is related to Loan Density   using Repayment Pattern Density  by the following Ordinary Differential Equation: with initial condition  RD (, ) =   () that expresses that The Current Debt at time  which is related to the borrowed amount at time  is the borrowed amount at time .The solution of this differential equation is expressed as We will use the expression (81) of density  in order to express Algebraic Spending Density  I in the model with variable rate.Quantity  I (, ) is defined such that the difference between spending and income at time  is associated with the amount borrowed at time .It is a time density with respect to both variables  and .Relation (81) yields where  I is the Interest Payment Density at time  which is associated with the borrowed amount at time , and where  I K is a repayment scheme at time  which is associated with the borrowed amount at time .
In what follows, we will show that the definition of Algebraic Spending Density  I is consistent with the definition of Algebraic Spending  which is given in relation (6).Indeed, Algebraic Spending Density  can be expressed in terms of  I .By integration over variable  (which describes the borrowed time), from  I (, ), Algebraic Spending Density  can be defined as follows: where density   is to be determined, which is Algebraic Spending Density associated with K RD ( I ) the known Current Debt at initial time  I .

𝜎 (𝑡
where density   can be expressed as follows: We will justify the expression (91) of density   as follows.Indeed, replacing time  by initial time  I in expressions ( 6) and ( 91 (92)

Inverse Problem of the Model in S(R + )
Denoting S(R + ) the Schwartz space consists of smooth functions whose derivatives (including the function itself) decay at positive infinity faster than any power.We say, for short, that Schwartz functions are rapidly decreasing.We state the Repayment Pattern Density  as follows: We use the Fourier Transform which are operators acting on densities over R. Operators F stand for the Fourier Transform, and F −1 stands for the Inverse Fourier Transform.
From this, we get the following equality: then, F(  ) ∈ L ∞ (R) and is such that If L[  ] does not satisfy the equality in relation (100), then, F(  ) has an infinite limit in 0.