# Some Notes about the Continuous-in-Time Financial Model

Authors: Tarik Chakkour
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

In this paper, we investigate the properties of operators in the continuous-in-time model which is designed to be used for the finances of public institutions. These operators are involved in the inverse problem of this model. We discuss this inverse problem in Schwartz space that we prove the uniqueness theorem.


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## Body

## 1. 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-in-time 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~E 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~E are absolutely continuous with respect to the Lebesgue measure dt. This means that they read σ(t)dt and κE(t)dt, where t is the variable in R. We call σ and κE time densities.LetF and G be normed spaces. Throughout this paper, L:F→G is a continuous linear application (in short, an operator). We say that the following problem: (1)Givenσ∈G,findκE∈Fsuchthatσ=LκE, is well-posed if L is invertible and its inverse L-1: G→F is continuous. In other words, the problem is said to be well-posed if(2)σ=LκE;∀σ∈G,∃!κE∈F(3)thesolutionκEdependscontinuouslyonσ.Existence and uniqueness of a solution for all g∈G (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 σ and Loan Density κE. 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~E from Algebraic Spending Measure σ~ in Radon measure space, that is, F=M([tI,Θmax-Θγ]) and G=M([tI,Θmax]), and in Hilbert space, that is, F=L2([tI,Θmax-Θγ]) and G=L2([tI,Θmax]), 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 operatorL, describing the computation of its image.The rest of this paper is arranged as follows. In Section2 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.


## 2. 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 overR. For that, we will also consider that M([tI,Θmax]) is the set of Radon Measures over R, supported in [tI,Θmax]. 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 denotingL2([tI,Θmax]) the space of square-integrable functions over R having their support in [tI,Θmax]. We state the Repayment Pattern Density γ as follows:(4)γ∈L20,Θγ,where Θγ is a positive number such that(5)Θγ<Θmax-tI.We recall that we have shown the balanced equation given by equality (14) in [2]. This equality consists in writing Loan Density κE as a sum of Algebraic Spending Density σ 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 σ:(6)σt=κEt-κE⋆γt-α∫tItκE-κE⋆γsds-α∫tΘmaxρKIsds-ρKIt.From this, linear term of density σ is defined by linear operator L acting on Loan Density κE∈L2([tI,Θmax-Θγ]) given by(7)LκEt=κEt-κE⋆γt-α∫tItκE-κE⋆γsds.The aim here is to compute operator L∗. For that, we will compute inner product 〈L[κE],κF〉 for any densities κE and κF in, respectively, spaces L2([tI,Θmax-Θγ]) and L2([tI,Θmax]) defined by(8)LκE,κF=∫tIΘmaxLκExκFxdx=∫tIΘmaxκExκFxdx-∫tIΘmaxκE⋆γxκFxdx-α∫tIΘmax∫tIxκEydyκFxdx+α∫tIΘmax∫tIxκE⋆γydyκFxdx.We will simplify inner product 〈L[κE],κF〉 given by relation (8) in function of four terms. Since first term ∫tIΘmaxκE(x)κF(x)dx is already simplified, we simplify the second one as follows:(9)∫tIΘmaxκE⋆γxκFxdx=∫tIΘmax∫tIxγx-yκEydyκFxdx=∫tIΘmax∫tIΘmaxγx-yκEy1y≤xdyκFxdx=∫tIΘmaxκEy∫yΘmaxγx-yκFxdxdy.Next, we simplify the third one as follows:(10)∫tIΘmax∫tIxκEydyκFxdx=∫tIΘmax∫tIΘmaxκEy1y≤xdyκFxdx=∫tIΘmaxκEy∫yΘmaxκFxdxdy.The last one is simplified as follows:(11)∫tIΘmax∫tIxκE⋆γydyκFxdx=∫tIΘmax∫tIx∫tIyγy-tκEtdtdyκFxdx=∫tIΘmax∫tIΘmax1y≤x∫tIΘmaxγy-tκEt1t≤ydtdyκFxdx=∫tIΘmax∫tIΘmaxκEt1t≤x∫tIΘmaxγy-t1t≤y≤xdydtκFxdx=∫tIΘmax∫tIΘmaxκFx1t≤x∫tIΘmaxγy-t1t≤y≤xdydxκEtdt.According to relations (9), (10), and (11), expression (8) of inner product 〈L[κE],κF〉 reads(12)LκE,κF=∫tIΘmaxκEyκFy-∫yΘmaxγx-yκFxdx-α∫yΘmaxκFxdx+α∫yΘmaxκFx∫yxγz-ydzdxdy.Since operator L∗ is one such that(13)LκE,κF=κE,L∗κF,∀κE∈L2tI,Θmax-Θγ,∀κF∈L2tI,Θmax,and, according to relation (12), operator L∗ is defined by(14)L∗κFy=κFy-∫yΘmaxγx-yκFxdx-α∫yΘmaxκFxdx+α∫yΘmaxκFx∫yxγz-ydzdx=κFy-∫yΘmaxκFxγx-y+α-α∫yxγz-ydzdx.Define linear operator D acting on Initial Debt Repayment Density ρKI∈L2([tI,Θmax]) as(15)DρKIt=-α∫tΘmaxρKIsds-ρKIt.The integration by parts states that inner product 〈D[ρKI],κF〉 is computed for any densities ρKI and κF in L2([tI,Θmax]) as follows:(16)DρKI,κF=∫tIΘmaxDρKIxκFxdx=-α∫tIΘmax∫xΘmaxρKIydyκFxdx-∫tIΘmaxρKIxκFxdx=α∫tIΘmaxρKIx∫xΘmaxκFydydx+α∫tIΘmaxρKIydy∫tIΘmaxκFydy-∫tIΘmaxρKIxκFxdx=∫tIΘmaxρKIxα∫xΘmaxκFydy+α∫tIΘmaxκFydy-κFxdx.From this, operator D∗ is defined by(17)D∗κFx=α∫xΘmaxκFydy+α∫tIΘmaxκFydy-κFx.Operator L we set out in relation (7) considered a constant rate. Nevertheless, if we consider in it a function α that depends on t, the model becomes a financial model with variable rate. The only modification to make is to enrich (7) and (15) by writing(18)LκEt=κEt-κE⋆γt-αt∫tItκE-κE⋆γsds.(19)DρKIt=-αt∫tΘmaxρKIsds-ρKIt.Once this enrichment is done, using a variable rate makes operator D∗ expressed in terms of densities α and γ:(20)L∗κFy=κFy-∫yΘmaxκFxγx-y+αx-αx∫yxγz-ydzdx.By definition α(x) is the rate at time x. Operator D∗ given by relation (17) can be rewritten adding this enrichment(21)D∗κFx=αx∫xΘmaxκFydy+αx∫tIΘmaxκFydy-κFx.Lemma 1.
The image of operatorL is such that(22)Im⁡L⊥=0.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:(23)Im⁡L⊥=Ker⁡L∗.According to (14), if density κF is in Ker⁡(L∗), then, we get the following equation:(24)κFy-∫yΘmaxκFxγx-y+α-α∫yxγz-ydzdx=0.Deriving, we get the ODE that the solution κF is expressed as follows:(25)κF′y-α+γ0κFy=0.The general solution to (25) is given by(26)κFy=κFtIe-α+γ0y-tI,where initial condition κF(tI) stands for the value of density κF at initial time tI. On the other hand, (24) is equivalent to(27)κFy-α+γ0∫yΘmaxκFxdx︸FirsttermI1+∫yΘmax-γx-y+γ0+α∫yxγz-ydzdx︸SecondtermI2=0.Conversely, we will show that density κF is zero. In the first place, replacing density κF given by (26) in first term I1 of (27), we get(28)κFy-α+γ0∫yΘmaxκFxdx=-ακFtIα+γ0e-α+γ0Θmax-tI.Secondly, the second term I2 is a constant function due to its derivative which equals zero:(29)∫yΘmax-γx-y+γ0+α∫yxγz-ydzdx′=0--γ0+γ0+α∫yyγz-ydz=0.Consequently, the second term I2 is equal to a real constant C to be determined:(30)∫yΘmax-γx-y+γ0+α∫yxγz-ydzdx=C.The initial condition is obtained from integral equation defined by (30) with replacing y by Θmax, which implies that constant C is zero. It is concluded that(31)∫yΘmax-γx-y+γ0+α∫yxγz-ydzdx=0,∀y∈tI,Θmax.Then, relations (27), (28), and (31) yield the following equality:(32)-ακFtIα+γ0e-α+γ0Θmax-tI=0.From this, initial density κF(tI) or loan rate α is zero since exponential function is positive. It follows that if κF(tI) is zero, then, density κF is zero, which is obtained from relation (26). In this case:(33)Ker⁡L∗=0.If loan rate α is zero, then, according to (24), we obtain following integral equation:(34)κFy-∫yΘmaxκFxγx-ydx=0,where expression of density κF is determined from equality (26) as(35)κFy=κFtIe-γ0y-tI.If density κF given by (35) is coupled with (34), then, κF(tI) is zero allowing zero density κF. We showed that density κF is zero in both cases. From this, we can deduce that (33) is true, proving the lemma.


## 3. 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 operatorV~:L2([tI,Θmax-Θγ])→L2([tI,Θmax]) by operator that is acting on Loan Density κE,(36)V~κEt=κE⋆γt+α∫tItκE-κE⋆γsds.The canonical injection IdL2([tI,Θmax-Θγ])→L2([tI,Θmax]) is defined from L2([tI,Θmax-Θγ]) to L2([tI,Θmax]) as(37)IdL2tI,Θmax-Θγ→L2tI,ΘmaxκEt=κEt,which is decomposed as a sum of operators L and V~ given by relations (7) and (36), respectively:(38)LκE+V~κE=IdL2tI,Θmax-Θγ→L2tI,Θmax.Theorem 2.
If densityγ has upper bound M-2|α| over its support:(39)supz∈0,Θγ⁡γz=M-2α,where M is a positive real satisfying(40)2α<M<1Θmax-tI;then, operator L is invertible, where its inverse L-1 is given by(41)L-1κEx=J∘∑k≥0V~∘JkκEx,where J is an operator defined by(42)J:L2tI,ΘmaxκE⟶⟼L2tI,Θmax-ΘγκEProof.
Since operatorV~ is defined from L2([tI,Θmax-Θγ]) to L2([tI,Θmax]) and by using definition (42) of operator J, we get(43)V~∘JL2tI,Θmax⊂L2tI,Θmax.From this, we get(44)V~∘JkL2tI,Θmax⊂L2tI,Θmax,∀k∈N,which implies that(45)J∘∑k≥0V~∘JkL2tI,Θmax⊂L2tI,Θmax-Θγ.Relation (45) shows that equality (41) is consistent due to the inverse of operator L being in L2([tI,Θmax-Θγ]). Next, we can show that operator V~∘J can be written in the form(46)V~∘JκEx=∫tIxkx,yκEydy,where kernel k is defined as(47)kx,y=γx-y+α-α∫yxγt-ydt.Following equality(48)y,t∈R2/tI≤y≤t,tI≤t≤x=y,t∈R2/tI≤y≤x,y≤t≤xyields with Fubini-Tonelli theorem to obtain(49)V~∘J2κEx=∫tIxkx,tV~∘JκEtdt=∫tIxkx,t∫tItkt,yκEydydt=∫tIx∫yxkx,tkt,ydtκEydy.Consequently, operator V~∘J(2) is written in following form:(50)V~∘J2κEx=∫tIxk2x,yκEydy,where(51)k2x,y=∫yxkx,tkt,ydt.We can verify by induction for n≥2 that the recurrence expresses each operator V~∘J(n) as an integral operator which is written in following form:(52)V~∘JnκEx=∫tIxknx,yκEydy,where kernel kn is given as(53)knx,y=∫yxkx,tkn-1t,ydt.Now, we will show that M is a maximum of kernel k over [tI,Θmax]×[tI,Θmax-Θγ] with using equality (39):(54)kx,y≤γx-y+α∫0Θγγzdt+α≤M-2α+α+α≤M. By a recurrence starting with initial state n=1 being true, it is easy to prove that(55)knx,y≤Mnx-yn-1n-1!.If n=1, inequality (55) is true because M is the maximum of the kernel k over [tI,Θmax]×[tI,Θmax-Θγ]. Assume that inequality (55) is true for a case n and show a case n+1. According to (53), we get(56)kn+1x,y≤∫yxkx,tknt,ydt≤M×Mnn-1!∫yxt-yn-1dt≤M×Mnn-1!×x-ynn≤Mn+1x-ynn!. Since x-y∈[0,Θγ], we use inequality (55) to get(57)knx,y≤MnΘγn-1n-1!.Applying Cauchy-Schwarz inequality at V~∘J(n) defined by (52), we get following equality:(58)V~∘JnκEx2≤∫tIxknx,y2dyκEL2tI,Θmax-Θγ2.From this and using inequality (57), we get(59)V~∘JnκEx2≤Mnn-1!2∫tIxx-y2n-1dyκEL2tI,Θmax-Θγ2≤Mnn-1!2×x-tI2n-12n-1κEL2tI,Θmax-Θγ2.By integrating each term of (60) over interval [tI,Θmax], we get(60)V~∘JnκEL2tI,Θmax2≤Mnn-1!2×Θmax-tI2n2n2n-1κEL2tI,Θmax-Θγ2. Consequently,(61)V~∘JnL2tI,Θmax≤Mnn-1!×Θmax-tIn2n. Inequality (61) gives(62)V~∘JnL2tI,Θmax1/n≤MΘmax-tI×2nn-1!-1/n. Since the series ∑k≥1ak of defined terms(63)ak=MΘmax-tIk2kk-1!is convergent, then quantity ∑k≥0V~∘J(k) converges absolutely in L2([tI,Θmax]). Consequently, ∑k≥0V~∘J(k) exists and is finite in L2([tI,Θmax]). We recall that we have(64)IdL2tI,Θmax-V~∘J∘∑k=0NV~∘JkκEx=∑k=0NV~∘JkκEx-∑k=0NV~∘Jk+1κEx=IdL2tI,ΘmaxκEx-V~N+1κEx. Since we have n≤n-1! for all integer n≥2, inequality (61) implies that(65)V~∘JnL2tI,Θmax≤MΘmax-tIn2n. Coupling inequality (64) with the fact that V~∘JnL2([tI,Θmax]) converges to 0 due to (40), we get(66)IdL2tI,Θmax-V~∘J∘∑k≥0V~∘JkκEx=IdL2tI,ΘmaxκEx.Composing operators defined by (38) with operator J, we get(67)L∘JκEx=IdL2tI,Θmax-Θγ→L2tI,Θmax∘JκEx-V~∘JκEx=IdL2tI,ΘmaxκEx-V~∘JκEx. According to (66) and (67), we get(68)L∘J∘∑k≥0V~∘JkκEx=IdL2tI,ΘmaxκEx.That is,(69)L∘J∘∑k≥0V~∘JkκEx=IdL2tI,ΘmaxκEx,achieving equality (41) of the lemma.Let us characterize under assumption of Theorem2 a spectrum set Sp(V~∘J) of operator V~∘J. Defining spectrum Sp(V~∘J) as a set of reals μ such that V~∘J-μIdL2([tI,Θmax]) is not invertible. Formally:(70)SpV~∘J=μ∈R:V~∘J-μIdL2tI,Θmaxisnotinvertible.Defining spectral radius ρ(V~∘J) of operator V~∘J as(71)ρV~∘J=λ:λ∈SpV~∘J.Note that spectrum Sp(V~∘J) is included in a disk of center 0 and radius V~∘JL2([tI,Θmax]). That is to say(72)ρV~∘J≤V~∘JL2tI,Θmax.Now we will compute spectral radius ρ(V~∘J) of operator V~∘J using following equality:(73)ρV~∘J=limn→+∞V~∘JnL2tI,Θmax1/n.Since we have shown that V~∘J(n)L2([tI,Θmax])1/n converges to 0, equality (73) shows that spectral radius ρ(V~∘J) is zero:(74)ρV~∘J=0.Consequently, spectrum Sp⁡V~∘J is reduced to null set:(75)Sp⁡V~∘J=0.


## 4. 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 σI in the model with variable rate. The mathematical consistency of this density σI is analyzed.The Current Debt FieldKRD is related to Loan Measure κ~E and Repayment Measure ρ~K by the following Ordinary Differential Equation:(76)dKRDdt=κEt-ρKt-ρKIt.The solution of this ODE is expressed as follows:(77)KRDt=KRDtI+∫tItκEsds-∫tItρKsds-∫tItρKIsds=∫tItκEsds-∫tItρKIsds+∫tΘmaxρKIsds. Since the Interest Payment Density ρI is related to the Current Debt Field by a proportionality relation:(78)ρIt=αtKRDt;the Interest Payment Density ρI can be expressed in terms of Loan Density κE:(79)ρIt=αt∫tItκEsds-αt∫tItκE⋆γsds+αt∫tΘmaxρKIsds.Since Density σ reads(80)σt=LκEt+DρKIt,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:(81)σt=dKRDtdt-ρIt.Defining kRD(t,s) the Current Debt at time t which is associated with the amount borrowed at time s. kRD is related to Loan Density κE using Repayment Pattern Density γ by the following Ordinary Differential Equation:(82)dkRDt,sdt=-γt-sκEs,with initial condition kRD(s,s)=κE(s) that expresses that The Current Debt at time s which is related to the borrowed amount at time s is the borrowed amount at time s. The solution of this differential equation is expressed as(83)kRDt,s=κEs-∫stγy-sκEsdy.We will use the expression (81) of density σ in order to express Algebraic Spending Density σI in the model with variable rate. Quantity σI(t,s) is defined such that the difference between spending and income at time t is associated with the amount borrowed at time s. It is a time density with respect to both variables s and t. Relation (81) yields(84)σIt,s=dkRDt,sdt+dkRDs,sds-rKIt,s-rIt,s,where rI is the Interest Payment Density at time t which is associated with the borrowed amount at time s, and where rKI is a repayment scheme at time t which is associated with the borrowed amount at time s.In what follows, we will show that the definition of Algebraic Spending DensityσI is consistent with the definition of Algebraic Spending σ which is given in relation (6). Indeed, Algebraic Spending Density σ can be expressed in terms of σI. By integration over variable s (which describes the borrowed time), from σI(t,s), Algebraic Spending Density σ can be defined as follows:(85)σt=∫tItσIt,sds+σIt,where density σI is to be determined, which is Algebraic Spending Density associated with KRD(tI) the known Current Debt at initial time tI.(86)σt=-∫tItγt-sκEsds+∫tItdκE-∫tItrKIt,sds-∫tItrIt,sds+σIt.Replacing first term in relation (86) by -κE⋆γ and using the fact that the Interest Payment Density ρI is expressed in [2] as follows:(87)rIt,s=αsκEs-αs∫stγy-sκEsdy,we get:(88)σt=κEt-κE⋆γt-κEtI-∫tItrKIt,sds-∫tItαsκEsds+∫tItαs∫stγy-sκEsdyds+σIt. The integration of Density rKI over variable s gives(89)ρKIt=∫tItrKIt,sds+ρKItI.Expression (88) and consequently definition of Borrowed Time Related Algebraic Spending Density σI are consistent with the expression of Algebraic Spending Density σ given in relation (6). Indeed, if the rate α(t) is fixed with worth α, (88) writes(90)σt=κEt-κE⋆γt-κEtI-∫tItrKIt,sds-α∫tItκEsds+α∫tIt∫stγy-sκEsdyds+σIt=κEt-κE⋆γt-ρKIt-α∫tItκEs-∫tIyγy-sκEsdsdy-κEtI+ρKItI+σIt=κEt-κE⋆γt-α∫tItκE-κE⋆γsds-ρKIt-κEtI+ρKItI+σIt, where density σI can be expressed as follows:(91)σIt=κEtI-ρKItI-α∫tΘmaxρKIsds.We will justify the expression (91) of density σI as follows. Indeed, replacing time t by initial time tI in expressions (6) and (91) of Density σ, we obtain the same expression σ(tI) defined by(92)σtI=κEtI-κE⋆γtI-ρKItI-α∫tIΘmaxρKIsds.


## 5. Inverse Problem of the Model inS(R+)

DenotingS(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 γ as follows:(93)γ∈SR+.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.Lemma 3.
If functionκE is in S(R+) and if γ satisfies relation (93), then we have the following equality:(94)1-FγFκE=FLκE+α∫tI∙LκEseα∙-sds.Proof.
Integrating by parts states that(95)∫tIt∫tIsκE-κE⋆γydy×αteαt-sds=∫tItκE-κE⋆γs×eαt-sds-∫tItκE-κE⋆γsds.From this, we get the following equality:(96)∫tItκE-κE⋆γsds=∫tItκE-κE⋆γs-αt∫tIsκE-κE⋆γydy×eαtt-sds.Using definition (18) of operator L, equality in (96) is multiplied by density α(t) to give(97)αt∫tItκE-κE⋆γsds=αt∫tItLκEseαt-sds.Replacing α(t)∫tIt(κE-κE⋆γ)(s)ds in relation (97) by (κE-κE⋆γ)(t)-L[κE](t) which is possible because of (18), we obtain the following equality:(98)κEt-κE⋆γt=LκEt+αt∫tItLκEseα∙-sds.Applying Fourier Transform to each term of equality (98), we obtain equality (94), proving the lemma.Lemma 4.
Assuming (93), that implies that(99)∫0+∞yγydy≠0is achieved and if function L[κE] given by relation (7) satisfies(100)∫0+∞LκEy+α∫0yLκEseαy-sdsdy=0and for a negative rate α(101)t⟼∫0tLκEsds∈SR+,then, F(κE)∈L∞(R) and is such that(102)limξ→0⁡FκEξ=-2Θγ2∫0+∞y×LκEy+α∫0yLκEseαy-sdsdy.If L[κE] does not satisfy the equality in relation (100), then, F(κE) has an infinite limit in 0.Proof.
AsL[κE]∈S(R+) and the fact that S(R+)⊂L1(R+), we get(103)LκE+α∫0∙LκEseα∙-sds∈L1R+.Indeed, under assumption (101) the product of a bounded function t↦eα(t-s) due to the negative rate α by a function in L1(R+) is a function in L1(R+). Then, using an order 1 Taylor expansion of e-iyξ, we obtain the following expansion of function F(L[κE]+α∫0∙L[κE](s)eα(∙-s)ds):(104)FLκE+α∫0∙LκEseα∙-sdsξ=∫0+∞LκEy+α∫tIyLκEseαy-sdsdy-iξ∫0+∞y×LκEy+α∫0yLκEseαy-sdsdy+Oξ2.Since operator L satisfies equality in relation (100), relation (104) gives(105)FLκE+α∫tI∙LκEseα∙-sdsξ=-iξ∫0+∞y×LκEy+α∫tIyLκEseαy-sdsdy+Oξ2.According to (93), function 1-F(γ) is Taylor expanded in 0 until the order 1 to obtain(106)1-Fγξ=iξ∫0+∞yγydy+Oξ2.According to relations (94), (105), and (106), we get equality (102). Moreover, according to relation (94), F(κE) is a finite quantity outside of 0. Hence, it is concluded that F(κE) is in L∞(R). On the other hand, if equality in relation (100) is not satisfied, then, according to relations (94), (104), and (106), F(κE) has an infinite limit at 0. From this, the proof of the lemma is achieved.Theorem 5.
If Repayment Pattern Densityγ satisfies relation (93) such that(107)Fγ<1and if Initial Debt Repayment Density ρKI is in S(R+), then for any Algebraic Spending Density σ in S(R+) satisfying the following equality:(108)∫0+∞σy-DρKIy+α∫0yσs-DρKIseαy-sdsdy=0,there exists an unique Loan Density κE in S(R+) which is given in terms of σ by(109)κE=F-1Fσ-DρKI+α∫0∙σs-DρKIseα∙-sds1-Fγ.Proof.
According to relation (107), we get(110)11-Fγ=∑k≥0Fkγ.In the first place, we will show that the left term given by (110) is in S(R+). Indeed, since S(R+) is stable by Fourier Transform, assumption (93) states that F(γ) is in S(R+). From this, quantity 1/1-F(γ) is in S(R+) due to its stability under power and infinite sum. Secondly, we will show that density σ-D[ρKI]+αF is in S(R+), where density F is defined by(111)Ft=∫0tσs-DρKIseαt-sds.Because densities σ and D[ρKI] are in S(R+), density σ-D[ρKI] is rapidly decreasing. Thus, for an integer N′ there exists a positive constant CN′ such that(112)σs-DρKIs≤CN′1+sN′,∀s∈R+.Since the derivative of density F is equal to σ-D[ρKI], inequality (112) gives(113)F′s≤CN′1+sN′,∀s∈R+.Integrate inequality (113) over [0,t] to get(114)Ft≤CN′N′-11-11+tN′-1,∀t∈R+.Consequently, for an integer N there exists a positive constant CN such that(115)Ft≤CN1+tN,∀t∈R+.Inequality (115) states that F is rapidly decreasing. Furthermore, since the (l+1)-nd derivative of density F is equal to the l-nd derivative of density σ-D[ρKI],(116)Fl+1t=σ-DρKIlt,∀l∈N,∀t∈R+,and the fact that space S(R+) is stable under the operation of derivation, for any integer l function F(l+1) is rapidly decreasing. Consequently, F is in S(R+), achieving the second point of proof. We conclude that(117)Fσ-DρKI+α∫0∙σs-DρKIseα∙-sds1-Fγ∈SR+,because it is the product of two functions in S(R+).



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